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I bookboon.com Advanced stochastic processes: Part I Jan A. Van Casteren Download free books at book boon. com Jan A. Van Casteren Advanced Stochastic Processes Part I ii Download free eBooks at bookboon.com Advanced stochastic processes: Part I 2 nd edition © 2015 Jan A. Van Casteren & bookboon.com ISBN 978-87-403-1115-0 The author is obliged to Department of Mathematics and Computer Science of the University of Antwerp for its material support. The author is also indebted to Freddy Delbaen (ETH, Zurich), and the late Jean Haezendock (University of Antwerp). A big part of Chapter 5 is due to these people. The author is thankful for comments and logistic support by numerous students who have taken courses based on this text. In particular, he is grateful to Lieven Smits and Johan Van Biesen (former students at the University of Antwerp) who wrote part of the Chapters 1 and 2. The author gratefully acknowledges World Scientic Publishers (Singapore) for their permission to publish the contents of Chapter 4 which also makes up a substantial portion of Chapter 1 in [144]. The author also learned a lot from the book by Stirzaker [124]. Section 1 of Chapter 2 is taken from [124], and the author is indebted to David Stirzaker to allow him to include this material in this book. The author is also grateful to the people of Bookboon, among whom Karin Hamilton Jakobsen and Ahmed Zsolt Dakroub, who assisted him in the final stages of the preparation of this book. iii Download free eBooks at bookboon.com Advanced stochastic processes: Part I Contents Contents Preface i Chapter 1. Stochastic processes: prerequisites 1 1. Conditional expectation 2 2. Lemma of Borel-Cantelli 9 3. Stochastic processes and projective systems of measures 10 4. A definition of Brownian motion 16 5. Martingales and related processes 17 Chapter 2. Renewal theory and Markov chains 35 1. Renewal theory 35 2. Some additional comments on Markov processes 61 3. More on Brownian motion 70 4. Gaussian vectors. 76 5. Radon-Nikodym Theorem 78 6. Some martingales 78 www.sylvania.com We do not reinvent the wheel we reinvent light. Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges. An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day. OSRAM SYLVAN!A Light is OSRAM 4 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Contents Chapter 3. An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 89 1. Gaussian processes 89 2. Brownian motion and related processes 98 3. Some results on Markov processes, on Feller semigroups and on the martingale problem 117 4. Martingales, submartingales, supermartingales and semimartingales 147 5. Regularity properties of stochastic processes 151 6. Stochastic integrals, Ito’s formula 162 7. Black-Scholes model 188 8. An Ornstein-Uhlenbeck process in higher dimensions 197 9. A version of Fernique’s theorem 221 10. Miscellaneous 223 Deloitte Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. Download free eBooks at bookboon.com Advanced stochastic processes: Part I Contents Chapter 4. Stochastic differential equations 243 1. Solutions to stochastic differential equations 243 2. A martingale representation theorem 272 3. Girsanov transformation 277 Chapter 5. Some related results 295 1. Fourier transforms 295 2. Convergence of positive measures 324 3. A taste of ergodic theory 340 4. Projective limits of probability distributions 357 5. Uniform integrability 369 6. Stochastic processes 373 7. Markov processes 399 8. The Doob-Meyer decomposition via Komlos theorem 409 Subjects for further research and presentations 423 Bibliography 425 Index 433 SIMPLY CLEVER SKODA We will turn your CV into an opportunity of a lifetime Do you like cars? Would you like to be a part of a successful brand? We will appreciate and reward both your enthusiasm and talent. Send us your CV. You will be surprised where it can take you. 6 Send us your CV on www.employerforlife.com Download free eBooks at bookboon.com Advanced stochastic processes: Part I Preface Preface This book deals with several aspects of stochastic process theory: Markov chains, renewal theory, Brownian motion, Brownian motion as a Gaussian pro¬ cess, Brownian motion as a Markov process, Brownian motion as a martingale, stochastic calculus, Ito’s formula, regularity properties, Feller-Dynkin semi¬ groups and (strong) Markov processes. Brownian motion can also be seen as limit of normalized random walks. Another feature of the book is a thorough discussion of the Doob-Meyer decomposition theorem. It also contains some features of stochastic differential equations and the Girsanov transformation. The first chapter (Chapter 1) contains a (gentle) introduction to the theory of stochastic processes. It is more or less required to understand the main part of the book, which consists of discrete (time) probability models (Chap¬ ter 2), of continuous time models, in casu Brownian motion, Chapter 3, and of certain aspects of stochastic differential equations and Girsanov’s transfor¬ mation (Chapter 4). In the final chapter (Chapter 5) a number of other, but related, issues are treated. Several of these topics are explicitly used in the main text (Fourier transforms of distributions, or characteristic functions of random vectors, Levy’s continuity theorem, Kolmogorov’s extension theorem, uniform integrability); some of them are treated, like the important Doob-Meyer decomposition theorem, but are not explicitly used. Of course Ito’s formula implies that a C 2 -function composed with a local semi-martingale is again a semi-martingale. The Doob-Meyer decomposition theorem yields that a sub¬ martingale of class (DL) is a semi-martingale. Section 1 of Chapter 5 contains several aspects of Fourier transforms of probability distributions (characteristic functions). Among other results Bochner’s theorem is treated here. Section 2 contains convergence properties of positive measures. Section 3 gives some results in ergodic theory, and gives the connection with the strong law of large numbers (SLLN). Section 4 gives a proof of Kolmogorov’s extension theorem (for a consistent family of probability measures on Polish spaces). In Section 5 the reader finds a short treatment of uniform integrable families of functions in an iP-space. For example Scheffe’s theorem is treated. Section 6 in Chapter 5 contains a precise description of the regularity properties (like almost sure right-continuity, almost sure existence of left limits) of stochastic processes like submartingales, Levy processes, and others; it also contains a proof of Doob’s maximal inequality for submartingales. Section 7 of the same chapter contains a description of Markov process theory starting from just one probability space instead of a whole family. The proof of the Doob-Meyer decompositon theorem is based on a result by Komlos: see Section 8. Throughout the book the reader will be exposed to martingales, and related processes. Download free eBooks at bookboon.com Advanced stochastic processes: Part I Preface Readership . From the description of the contents it is clear that the text is designed for students at the graduate or master level. The author believes that also Ph.D. students, and even researchers, might benefit from these notes. The reader is introduced in the following topics: Markov processes, Brownian motion and other Gaussian processes, martingale techniques, stochastic differ¬ ential equations, Markov chains and renewal theory, ergodic theory and limit theorems. ii Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites CHAPTER 1 Stochastic processes: prerequisites In this chapter we discuss a number of relevant notions related to the theory of stochastic processes. Topics include conditional expectation, distribution of Brownian motion, elements of Markov processes, and martingales. For com¬ pleteness we insert the definitions of a a-field or cx-algebra, and concepts related to measures. 1.1. Definition. A cx-algebra, or cx-field, on a set is a subset A of the power set IP (Q) with the following properties: (i) £ A; (ii) A e A implies A c : = U\A e A; 00 (iii) if {A n ) n>1 is a sequence in A, then A n belongs to A. n — 1 Let A be a u-field on Q. Unless otherwise specified, a measure is an application p, : A —> [0, oo] with the following properties: • h (0) = 0; • if (A n ) n>l is a mutually disjoint sequence in A, then ( oo \ N oo (jA n j= p (A n ) = p (A n ). n —1 / iV— kx)j = 1 n — 1 If /./ is measure on A for which // (Q) = 1, then /.i is called a probability measure; if n (fl) ^ 1, then p is called a sub-probability measure. If fi : A —*■ [0,1] is a probability space, then the triple (U, A, fi) is called a probability space, and the elements of A are called events. Let M be a collection of subsets of CP (Q), where Q is some set like in Definition 1.1. The smallest a-field containing M is called the cr-field generated by M, and it is often denoted by a (M). Let (U, A,/i) be a sub-probability space, i.e. ji is a sub-probability on the a-field A. Then, we enlarge 0 with one point A, and enlarge A to A a := ct (A u {A}) = {A e CP (U A ) : A n f2 e A} . Then /r A : A a —> [0,1], defined by /r A (A) = ju (A n f2) + (1 — /i (U)) 1^ (A) , AgA a , (1.1) turns the space (0 A ,A A ,/i A ) into a probability space. Here fl A = Slu {A}. This kind of construction also occurs in the context of Markov processes with 1 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites finite lifetime: see the equality (3.75) in (an outline of) the proof of Theorem 3.37. For the important relationship between Dynkin systems, or A-systems, and cr-algebras, see Theorem 2.42. 1. Conditional expectation 1.2. Definition. Let (fi,Al,P) be a probability space, and let A and B be P (A n B) events in A such that P [ B] > 0. The quantity P (A | B) = — ^ - is then called the conditional probability of the event A with respect to the event B. We put P (A | B) = 0 if P ( B ) = 0. Consider a finite partition {B 1; ..., B n } of tt with Bj e A for all j = 1,..., n , and let ‘B be the subfield of A generated by the partition {B L ,..., B n }, and write n V[A\H}-Y 1 V(A\B j )l Bl . 3 = 1 Then P \A | !B] is a “B-measurable stochastic variable on Q. and f F[A\‘B]dF= f l A dF for all B e £. Jb jb MAERSK I joined MITAS because I wanted real responsibility The Graduate Programme for Engineers and Geoscientists www.discovermitas.com Real work International opportunities Three work placements a I was a construction supervisor in the North Sea advising and helping foremen solve problems 2 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites Conversely, if / is a B-measurable stochastic variable on 11 with the property that for all B e B the equality fdP = 1, 4 dP holds, then f = P [-4 | B] P-almost surely. This is true, because (/ — P [A | B]) dP = 0 for all B e B. If B is a sub-field (more precisely a sub-cr-field, or sub- a- algebra) generated by a finite partition of Q. then for every A e A there exists one and only one class of variables in L 1 (Q, B, P), which we denote by P [A | B], with the following property I P [A | B] dP = I l A dP for all Be B. JB JB The variable Xp=i ^ iA \ Bj) Ir, is an element from the class P [^4 | B]. If we fix B e A with P ( B ) > 0, then the measure A >—> P (A | B) is a probability measure on (f2, A). If P ( B ) = 0, then the measure A >—> P (A | B) is the zero- measure. Let X be a P-integrable real or complex valued stochastic variable on Cl. Then X is also P (• | B)-integrable, and J XdP (- 1 B) = E ^ b] , provided P (B) > 0. This quantity is the average of the stochastic variable over the event B. As before, it is easy to show that if B is a subfield of A generated by a finite partition {Bi,...,B n } of Q, then there exists, for every P-integrable real or complex valued stochastic variable X on fl one and only one class of functions in L 1 (f2, B, P), which we denote by E [X | B] with the property that f E [X | B] dP = f XdP for all B e B. JB JB The variable 2y=i $ATdP (• | Bj) 1b 3 is an element from the class E [X | B]. The next theorem generalizes the previous properties to an arbitrary subfield (or more precisely sub-a-field) B of A. 1.3. Theorem (Theorem and definition). Let (f2,A,P) be a probability space and let B be a subfield of A. Then for every stochastic variable XeL 1 (0, A, P) there exists one and only one class in L 1 (fi, B, P), which is denoted by E [X | B] and which is called the conditional expectation of X with respect to B, with the property that f E [X | B] dP = f XdP for all BeB. JB JB If X = 1^, with A e A, then we write P [A | B] instead of E [l^ | B] ; if B is generated by just one stochastic variable Y , then we write E [X | T] and P [A | y] instead of respectively E [X | a (y)] and P [A | a (y)]. Proof. Suppose that X is real-valued; if X = Re X + ilm X is complex¬ valued, then we apply the following arguments to Re X and Im X. Upon writ¬ ing the real-valued stochastic variable X as X = X + — X~, where X ± are 3 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites non-negative stochastic variables in L 1 (0. A. P), without loss of generality we may and do assume that X ^ 0. Define the measure /j : A —» [0, go) by n(A) = XdF , A e A. Then /i is finite measure which is absolutely contin- Ja uous with respect to the measure P. We restrict // to the measurable space (f2, B); its absolute continuity with respect to P confined to (f2, B) is preserved. From the Radon-Nikodym theorem it follows that there exists a unique class Y e L 1 2 (0. B,P) such that, for all Be B, the following equality is valid: p(B) = I YdF, and hence | XdF = I YdF. Jb Jb Jb This proves Theoreml.3. □ If B is generated by a countable or finite partition {Bj : j e N}, then it is fairly easy to give an explicit formula for the conditional expectation of a stochastic X e L l (fil,A,F) with respect to B: k *<& E \X I Si = V--—- 1 1 J U p w) 1b,. jeN Next let B be an arbitrary subfield of A, let X belong to L 1 (fi, A.F), and let B be an atom in B. The latter means that P (B) > 0, and if A e B is such that A cz B, then either P (A) = 0 or P (B\A) = 0. If Y represents E [X | B], then YIb = bis, P-almost surely, for some constant b. This follows from the B-measurability of the variable Y together with the fact that B is an atom for (0, B, P). So we get $ B XdF = E [X | B] dF = J Yl B dF = bF (B), and hence b = P(B) Consequently, on the atom B we have: r . , LM E \X Bl = -^—r-zrr— = b , 111 PB P-almost surely. In particular, for X = 1 a, we have on the atom B the equality P(AnB) F[A\‘B] = P (B) P-almost surely. If B is not an atom, then the conditional expectation on B need not be constant. In the following theorem we collect some properties of conditional expectation. For the notion of uniform integrability see Section 5. 1.4. Theorem. Let (Q,.A, P) be a probability space, and let B be a subfield of A. Then the following assertions hold. (1) If all events in B have probability 0 or 1 (in particular if B is the trivial field {0,Q}), then for all stochastic variables X e L 1 (Q,.A,P) the equality E [X | B] = E (X) is true P -almost surely. (2) If X is a stochastic variable in L 1 (fl,/l,P) such that B and cr(X) are independent, then the equality E [X | B] = E (X) is true P -almost surely. 4 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites (3) If a and b are real or complex constants, and if the stochastic variables X and Y belong to L 1 (14, .A, P), then the equality E [aX + bY | B] = oE [X | B] + ME [Y \ B] is true P -almost surely. (4) If X and Y are real stochastic variables in L l (14, A., P) such that X < Y, then the inequality E [X | B] < E [F | B] holds P -almost surely. Hence the mapping X >—► E [X | B] is a mapping from L 1 (14, A, P) onto L 1 (14, B, P). (5) (a) If (X n : n e N) is a non-decreasing sequence of stochastic variables in L 1 (14, A , P), then sup E [X n | B] = E sup X n | B n n P-a/most surely. (b) If (X n : n e N) is any sequence of stochastic variables in L 1 (14, A, P) which converges P -almost surely to a stochastic vari¬ able X, and if there exists a stochastic variable Y e L 1 (14, A, P) such that \X n \ ^ Y for all neN, then lim E |X I Bl n—»oo L 1 J E lim X n _n—► oo P -almost surely, and in L 1 (14, B, P). The condition ‘\X n \ < Y for all n e N with Y e L 1 (14,/l,P)” may be replaced with “the sequence (X n ) neN is uniformly integrable in the space L 1 (f2,.A, P) ” and still keep the second conclusion in (5b). In order to have P -almost sure convergence the uniform integrability condition should be replaced with the condition ^ in£. ^supE [|X n |, |X n | > M | B] = 0, P -almost surely. (1.2) (6) If c(x) is a convex continuous function from M to M, and if X belongs to L 1 (14,X,P), then c (E [X | B]) < E [c(X) | B] , P -almost surely. (7) Let p ^ 1, and let X be a stochastic variable in L p P). Then the stochastic variable E [X | B] belongs to L p (Q, B,P), and ||E [X | B]|| p < ll^llp ■ So the linear mapping X i— > E [X | B] is a projection from LL (14, A , P) onto LL (14, B, P). (8) (Tower property) Let B' be another subfield of A such that B c B' c A. If X belongs to L 1 (14, X, P), then the equality E [E [X | B'] | B] = E [X | B] holds P -almost surely. (9) If X belongs to L 1 (14,B,P), then E [X | B] = X, P -almost surely. (10) If X belongs to L 1 (14, X, P), and if Z belongs to L 00 (14,B,P), then E [ZX | B] = BE [X | B] , P -almost surely. (11) If X belongs to L 2 (14,X,P), then E [Y (X - E (X | B])] = 0 for all Y e L 2 (14, B,P). Hence, the mapping X E [X | B] is an orthogonal projection from L 2 (14, A, P) onto L 2 (14, B, P). 5 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites Observe that for B the trivial cx-field, i.e. B = {0, 0}, the condition in (1.2) is the same as saying that the sequence (X n ) n is uniformly integrable in the sense that inf supETlXJ , \X n \ > Ml = 0. (1.3) Proof. We successively prove the items in Theorem 1.4. (1) For every Be '.B we have to verify the equality: I XdP = I E (X)tflP. Jb Jb If P (B) = 0, then both members are 0; if P ( B) = 1, then both mem¬ bers are equal to E(X). This proves that the constant E (X) can be identified with the class E [X | B]. (2) For every B e B we again have to verify the equality: XdP = E (X) dP. Employing the independence of X and B e B this can be seen as follows: f X dP = f X1 B dP = E [X1 B ] = E [X] E [1 b ] = f E [X] dP. (1.4) Jb Jq Jb Because achieving your dreams is your greatest challenge. IE Business School’s Master in Management taught in English, Spanish or bilingually, trains young high performance professionals at the beginning of their career through an innovative and stimulating program that will help them reach their full potential. Choose your area of specialization. Customize your master through the different options offered. Global Immersion Weeks in locations such as London, Silicon Valley or Shanghai. Because you change , we change with you . www.ie.edu/master-management mim.admissions@ie.edu f # In YwTube ii Master in Management • 6 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites (3) This assertion is clear. (4) This assertion is clear. (5) (a) For all B e “B and neNwe have E \X n | B] dF = X n dF. By (4) we see that the sequence of conditional expectations E [X n | , neN, increases P-almost surely. The assertion in (5a) then follows from the monotone convergence theorem. (b) Put X* = sup k>n X k , X** = inf k ^ n X k . Then we have — Y < X** < X n < X* < Y, P-almost surely. Moreover, the sequences (Y — X*) ngN and (Y + -T**) neN are increasing sequences consisting of non-negative stochastic variables with Y — lim sup n ^ 00 X n and Y + lim inf n ^.oo X n as their respective suprema. Since the sequence (X n ) ngN converges P-almost surely to X, it follows by (5a) together with (4) that E [X** | ®] t E [X** | ®] and E [X* \ 3] j E [X** \ T>]. 7 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites From the pointwise inequalities X** < X n < X* it then follows that lim E [X n 1B] = E [X | B], P-almost surely. Next let the uniformly integrable sequence ( X n ) n in L 1 (Q, A, P) be pointwise convergent to X. Then lim E[|X n — X|] = 0. What we need is n—>00 that lim E \\X n — XI I 23] = 0. (1.5) n^oo L i J Under the extra hypothesis (1.2) this can be achieved as follows: lim sup E [|X n — X| | B] n—>00 < lim sup E [|X n — X|, \X n — X\ < M | B] n^cc + lim sup E [\X n - X, \X n - X\ > M\ | B] n—>oo (apply what already has been proved in (5b), with |X n — X| in¬ stead of X n , to the first term) < lim sup E [|X n - X, \X n -X\> M\ | B]. (1.6) n—> oo In (1.6) we let M tend to oo, and employ (1.2) to conclude (1.5). This completes the proof of item (5). (6) Write c(x ) as a countable supremum of affine functions c(x) = sup L n (x), (1.7) neN where L n (z) = a n z + b n ^ c(z), for all those z for which c(z) < oo, i.e. for appropriate constants a n and b n . Every stochastic variable L n (X) is integrable; by linearity (see (3)) we have L n (E [X | B]) = E [L„ (X) | B], Hence L n (E[X | B]) < E[c(X) | B] . Consequently, c (E [X | B]) = sup L n (E [X | B]) < E [c(X) | B] . neN The fact that convex function can be written in the form (1.7) can be found in most books on convex analysis; see e.g. Chapter 3 in [28]. (7) It suffices to apply item (6) to the function c(x) = \x\ p . (8) This assertion is clear. (9) This assertion is also obvious. (10) This assertion is evident if Z is a finite linear combination of indicator functions of events taken from The general case follows via a limiting procedure. (11) This assertion is clear if Y is a finite linear combination of indicator functions of events taken from The general case follows via a limiting procedure. The proof of Theorem 1.4 is now complete. □ 8 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites 2. Lemma of Borel-Cantelli 1.5. Definition. The limes superior or upper-limit of a sequence (A n ) neN in a universe is the set A of those elements w e 0 with the property that oj belongs to infinitely many A n ’s. In a formula: A = lim sup A n = P| (J A k . n—>0 ° neN k^n The indicator-function 1 a of the limes-superior of the sequence (A n ) n6N is equal to the lim sup of the sequence of its indicator-functions: 1 a = lim sup 1 A n ■ n —>oo The limes inferior or lower-limit of a sequence (A n ) neN in a universe Q is the set A of those elements oj e Q with the property that, up to finitely many A As, the element (sample) oj belongs to all A n ’s. In a formula: A = lim inf A n = [ I P) A k . n —>00 1 1 neN k^n The indicator-function T 4 of the limes-inferior of the sequence (A n ) neN is equal to the lim inf of the sequence of its indicator-functions: 1 a = lim inf 1 A n - 71—>00 "I studied English for 16 years but... ...I finally learned to speak it in just n six lessons Jane, Chinese architect ENGLISH OUT THERE Click to hear me talking before and after my unique course download 1 Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites 1.6. Lemma. Let (ct n ) nEN be a sequence of real numbers such that 0 ^ a n < 1. Then lim n ^oo YIk=i a k < 00 */ and onl V */lim n _oo flLi (1 “ a k) > 0. Proof. For 0 < a < 1 the following elementary inequalities hold: -—— < log (1 — a) < —a. 1 — a Hence we see n -Z 1 — a k < log rid-«*> <k =1 n < - 2 «*• fc=l The assertion in Lemma 1.6 easily follows from these inequalities. □ 1.7. Lemma (Lemma of Borel-Cantelli). Let (A n ) ngM be a sequence of events, and put A = limsup^^ A n = f| n6 N U k>n A k- (i) If En=i P (An) < 00, then P (A) = 0. (ii) If the events A n , neN, are mutually ¥-independent, then the converse statement is true as well: P (A) < 1 implies Yjk=i P {Ak) < G0 > and hence X!*°=i P (A) = go if and only if P ( A ) = 1. PROOF, (i) For P (74) we have the following estimate: 00 P 0)«infX;P0A (1.8) k—n Since 2n=i P (A) < °o, we see that the right-hand side of (1.8) is 0. (ii) The statement in assertion (ii) is trivial if for infinitely many numbers k the equality P (A k ) = 1 holds. So we may assume that for all k e hi the probability P (Ak) is strictly less than 1. Apply Lemma 1.6 with a k = P (A k ) to obtain that X£Li P (Ak) < oo if and only if n n 0 < lim Ff (1 - P (A k )) = lim Ff P (0\A k ) n—>oo A n—>00 A k =1 k=l (the events (A k ) nen are independent) a)=1-P(A). (1.9) This proves assertion (ii) of Lemma 1.7. □ = lim P n—>oo <k =1 = lim P n—>oo f!\U k =1 3. Stochastic processes and projective systems of measures 1.8. Definition. Consider a probability space (0,A,P) and an index set I. Suppose that for every tel a measurable space (E t , £ t ) and an A-£ t -measurable mapping X(t) : Q —> E t are given. Such a family {X(t) : t e I) is called a stochastic process. 10 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites 1.9. Remark. The space 0 is often called the sample path space, the space E t is often called the state space of the state variable X(t). The a-field A is often replaced with (some completion of) the ex-field generated by the state variables X(t),t e I. This a-field is written as T. Let (S', S) be some measurable space. An T-S measurable mapping Y : Q —*■ S is called an S'-valued stochastic variable. Very often the state spaces are the same, i.e. (E t , £*) = (E. £), for all state variables X(t), tel. In applications the index set I is often interpreted as the time set. So / can be a finite index set, e.g. / = { 0 , 1 ,..., n], or an infinite discrete time set, like / = N = {0,1,...} or / = Z. The set I can also be a continuous time set: / = E or I = E + = [ 0 , go). In the present text, most of the time we will consider I = [ 0 , go). Let / be N, Z, E, or [0,go). In the so-called time-homogeneous or stationary case we also consider mappings —*• fl, s e I, s ^ 0 , such that X(t) ot} s = X(t + s), E-almost surely. It follows that these translation mappings : fl —» fl, s e I, are Sy-Tt-s-measurable, for all t ^ s. If Y is a stochastic variable, then Y o t) s is measurable with respect to the cx-field a {X(t) : t^s). The concept of time-homogeneity of the process (X(t) : tel ) can be explained as follows. Let Y : fl —> E be a stochastic variable; e.g. Y = nj-i/;(*(*;)), where fj : E —*• E, 1 ^ j ^ n, are bounded measurable functions. Define the transition probability P (s, B ) as follows: P (s, B) = P (X(s) e B), s e I, B e £. The measure B >—> E [Y o $ s , X(s) e B] is absolutely continuous with respect to the measure B >—> P (s, B), B e £. It follows that there exists a function F(s, x), called the Radon-Nikodym derivative of the measure B >—► E [Y o i) s . X(s) e B] with respect B >—► P (s, B ), such that E \Y o $ s , I(s)eB] = (s, x) P (s, dx). The function F (s,x) is usually written as F(s,x) = E[Vo-d s | X(s) e dx] E[Vod s , X(s) e dx] P[A(s) e dx] 1.10. Definition. The process (X(t) : tel ) is called time-homogeneous or stationary in time, provided that for all bounded stochastic variables Y : fl —> E the function E[Vod s | X(s) e dx] is independent of s e I, s ^ 0. In practice we only have to verify the property in Definition 1.10 for Y of the form Y = , where fj : E tj —*• E, 1 ^ j < n, are bounded measurable functions. Then Y o t) s = YYj=i fj (tj + s ))- This statement is a consequence of the monotone class theorem. 3.1. Finite dimensional distributions. As above let (Q, A, E) be a prob¬ ability space and let {X(t) : t e I] be a stochastic process where each state variable X(t) has state space (. E t , £ t ). For every non-empty subset J of I we write E J = YiteJ an d £ J = ® te j£ # denotes the product-field. We also write Xj = <g)tejX t . So that, if J = {t x ,..., t n ], then Xj = (X (E) ,...,X (t n )). The mapping Xj is the product mapping from fl to E J . The mapping Xj : fl —> E J 11 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites is £l-£ J -measurable. We can use it to define the image measure Pj: Pj (B) = XjP (. B ) = P [. Xj l B] = P[wef!: Xj(cj) e B] = P [Xj e B ], where B e £ J . Between the different probability spaces (E J , £ J ,Pj) there exist relatively simple relationships. Let J and H be non-empty subsets of I such that J a H, and consider the £' f/ -£ J -measurable projection mapping pj : E H —> E J , which ’’forgets” the ’’coordinates” in H\J. If H = /, then we write pj = pj. For every pair J and H with .J H c I we have Xj = pj o Xjj, and hence we get Pj (B) = PjFh ( B) = P h e B\, where B belongs to £ J . In particular if H = /, then P j (B) = pjF (B) = P [pj e B\, where B belongs to £j. If J = {t\,... ,t n } is a finite set, then we have Pj [Bi X • • • X B n ] = P [XJ 1 (Bi x • • • x B n )\ = P [X (ti) e Bi ,..., X ( t n ) e S n ], with e for 1 ^ j ^ n. 1.11. Remark. If the process {X(£) : t e 1} is interpreted as the movement of a particle, which at time £ happens to be in the state spaces E t) and if J = {ti,..., £ n } is a finite subset of /, then the probability measure Pj has the following interpretation: For every collection of sets B\ e £ tl ,..., B n e £ tn the number Pj [Si X • • • X £ n ] is the probability that at time t\ the particle is in £>i, at time 1 2 it is in £> 2 , • •and at time t n it is in B n . AACSB ACCREDITED Excellent Economics and Business programmes university of groningen www.rug.nl/feb/education uMSr p W “The perfect start of a successful, international career.” CLICK HERE to discover why both socially and academically the University of Groningen is one of the best places for a student to be 12 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites 1.12. Definition. Let fit be the collection of all finite subsets of I. Then the family { (E J , £ J . Pj) : J e fit} is called the family of finite-dimensional distri¬ butions of the process {X(t) : tel}] the one-dimensional distributions { (Et, £t, Pp}) : t 6 /} are often called the marginals of the process. The family of finite-dimensional distributions is a projective or consistent family in the sense as explained in the following definition. 1.13. Definition. A family of probability spaces {(E J , £ J ,Pj) : J e fit} is called a projective , a consistent system, or a cylindrical measure provided that Pj(B)=p* (P H ) (B) = P H [p* e B] for all finite subsets J cz H , J , H e fit, and for all sets B e £ J . 1.14. Theorem (Theorem of Kolmogorov). Let {(E J , £ J ,Pj) : J e fit} be a projective system of probability spaces. Suppose that every space E t is a cr - compact metrizable Hausdorff space. Then there exists a unique probability space (E 1 ,8, 1 with the property that for all finite subsets J e fit the equality Pj (L>) = P/ [pj 6 B] holds for all B s £ J . Theorem 5.81 is the same as Theorem 1.14, but formulated for Polish and Souslin spaces; its proof can be found in Chapter 5. Theorem 1.14 is the same as Theorem 3.1. The reason that the conclusion in Theorem 1.14 holds for o- compact metrizable topological Hausdorff spaces is the fact that a finite Borel measure p on a metrizable cr-compact space E is regular in the sense that ji(B) = sup p(K) = inf p(U), B any Borel subset E. (1.10) KczB, K compact U^>K, U open 1.15. Lemma. Let E be a a-compact metrizable Hausdorff space. Then the equality in (1.10) holds for all Borel subsets B of E. Proof. The equalities in (1.10) can be deduced by proving that the collec¬ tion T> define by T) = \ B e T> e : sup p{K) = inf p(U) KczB U=>B = \ B e T>e ■ sup p(F) = inf p(U) FczB U^B ( 1 . 11 ) contains the open subsets of E, is closed under taking complements, and is closed under taking mutually disjoint countable unions. The second equality holds because every closed subset of Li is a countable union of compact subsets. In (1.11) the sets K are taken from the compact subsets, the sets U from the open subsets, and the sets F from the closed subsets of Li. It is clear that T> is closed under taking complements. Let (x,y) d(x,y ) be a metric on E which 13 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites is compatible with its topology. Let F be a closed subset of E , and define U n by U n = \ x e E : inf d (x, y) < — > . ( yeF n) Then the subset U n is open, U n+ 1 => U n , and F = f)U n . It follows that // (F) = inf n /i (U n ), and consequently, F belongs to T). In other words the collection T contains the closed, and so the open subsets of E. Next let (B n ) n be a sequence of subsets in T. Fix £ > 0, and choose closed subsets F n a B n , and open subsets U n xj B n , such that fi (B n \F n ) ^ e2 n \ and /j,(U n \B n )^e 2 n . From (1.12) it follows that <: Uc„ \ n=l / \n=l oc vn=l n=l n=l From (1.13 it follows that = inf P<C/) : c/ => [J B n , U openj . The same argumentation shows that ( /oc \ /oc \\ OC 00 U B " \ U F d H S ^ e 2 2_ ” _1 - 2 e - \n=l / \n=l / / n=l n=l From (1.15) it follows that for N £ large enough. From (1.16 it follows that ^(Q Sn ) =sup j^ F): Fc u B n , F closed j- . ( 1 . 12 ) (1.13) (1.14) (1.15) (1.16) (1.17) From (1.14) and (1.17) it follows that (Jn=i belongs to T>. As already men¬ tioned, since every closed subset is the countable union of compact subsets the supremum over closed subsets in (1.17) may replaced with a supremum over compact subsets. Altogether, this completes the proof of Lemma 1.15. □ It is a nice observation that a locally compact Hausdorff space is metrizable and a -compact if and only if it is a Polish space. This is part of Theorem 5.3 (page 29) in Kechris [68]. This theorem reads as follows. 1.16. Theorem. Let E be a locally compact Hausdorff space. The following assertions are equivalent: 14 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites (1) The space E is second countable, i.e. E has a countable basis for its topology. (2) The space E is metrizable and a-compact. (3) The space E has a metrizable one-point compactification (or Alexan¬ dra ff compactification). (4) The space E is Polish, i.e. E is complete metrizable and separable. (5) The space E is homeomorphic to an open subset of a compact metrizable space. A second-countable locally-compact Hausdorff space is Polish: let (Ui) i be a countable basis of open subsets with compact closures ( K i ) i , and let V, be an open subset with compact closure and containing Ki. From Urysohn’s Lemma, let 0 < fi < 1 be continuous functions identically 0 off ly, identically 1 on Ki, and put d ( x , y ) = 2 * | fi ( x ) - fi ( y )\ + 2 = 1 E£i2 sii2 -*m x, y e E. (1.18) The triangle inequality for the usual absolute value shows that this is a metric. This metric gives the same topology, and it is straightforward to verify its completeness. For this argument see Garrett [57]. American online LIGS University is currently enrolling in the Interactive Online BBA, MBA, MSc, DBA and PhD programs: ► enroll by September 30th, 2014 and ► save up to 16% on the tuition! ► pay in 10 installments/2 years ► Interactive Online education ► visit www.ligsuniversity.com to find out more! Note: LIGS University is not accredited by nationally recognized accrediting agency by the US Secretary of Education. More info here. 15 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites 4. A definition of Brownian motion In this section we give a (preliminary) definition of Brownian motion. 4.1. Gaussian measures on M'h For every t > 0 we define the Gaussian kernel on W l as the function Pd(t,x,y ) 1 (2nt) d/2 2 1 J ' Then we have \pd (t, x, z ) dz = 1, and Pd(s, x, z)p(t, z, y) = p d (s + t, x, y)p d st sx + ty s T t s T t Hence the function pd (t, x, y) satisfies the equation of Chapman-Kolmogorov: J p d (s, x, z)p d (t, z, y)dz =Pd(s + t, x, y). This property will enable us to consider d-dimensional Brownian motion as a Markov process. Next we calculate the finite-dimensional distributions of the Brownian motion. 4.2. Finite dimensional distributions of Brownian motion. Let 0 < t\ < ■ ■ ■ < t n < co be a sequence of time instances in (0, co), and fix xo e W 1 . times) by (to = 0) Pxo;ti,...,tn [-®1 x ‘ ‘ ‘ x e IPxojti, t on the Borel r r n f dx n n i—i h ^3 >B n 3 = 1 j- 1 , Xj- 1 , Xj) , (1.19) where B \,..., B n are Borel subsets of R . Then, with B & = R , we have ■PiCo;^lv5^/s-l5^jt5^fc+lr--5^n [^1 ^ ^ ^k — 1 ^ ^ ^k +1 X * ' ' X dx n ... dxk+i dxk dxk-i •.. dx\ r r / JB i JBk—i Jb^+i JB k -1 j "J Pd (tj ~ tj- 1 , Xj- 1 , Xj) 3 =1 n Pd {tk ^k— 1 5 %k— 1 5 %k) Pd (tk+1 ^k-) Xfo-) Xk+ 1) P (tj tj—\,Xj—\,Xj) j=k +2 (Chapman-Kolmogorov) J • • • I I • • • I dx n ... dxk+i dx^—i ... dx\ Pd (tj tj— i, Xj_i, Xj) JBi JBk —i J-Bfc+i J-E?n 4=i Pd (tk+1 tfe— i, X^_i, J P (tj tj — i , Xj—i , Xj ) j=&+2 16 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites = [BiX ■■■ x B k _i x B k+1 x ■ ■ • x B n ]. (1.20) It follows that the family < | x • - x R^ , 23 d ® — ® 23 d ,Pa; 0 ; tl) ... )tn J;0 < ti < • • • < t n < oo, n e N >■ ^ \ n times n times / J is a projective or consistent system. Such families are also called cylindrical measures. The extension theorem of Kolmogorov implies that in the present situation a cylindrical measure can be considered as a genuine measure on the product field of Cl : = (R d ) L°' x \ This is the measure corresponding to Brownian motion starting at x 0 . More precisely, the theorem of Kolmogorov says that there exists a probability space (fi,T, P Xo ) and state variables X(t) : Q —> R d , t ^ 0, such that P xo [X (ti) eB u ...,X (t n ) e B n ] = P xo;tl ,..., tll [ B i x • • • x B n ], where the subsets Bj, 1 < j < n, belong to 23 d . It is assumed that P xo [A(0) = x 0 ] = l. 5. Martingales and related processes Let (Q, T, P) be a probability space, and let {dy : t e I } be a family of subfields of T, indexed by a totally ordered index set (/, <). Suppose that the family {T t : tel} is increasing in the sense that s ^ t implies T s . <= 'J t . Such a family of cr-fields is called a filtration. A stochastic process {X(t) : tel}, where X(t), tel, are mappings from fl to E t , is called adapted, or more precisely, adapted to the filtration {T* : tel} if every X(t) is T t -£rmeasurable. For the cr-field Bt we often take (some completion of) the cr-field generated by X(s), s < t: = o {A(s) : s ^ t}. 1.17. Definition. An adapted process {X(t) : t e 1} with state space (R, 23) is called a super-martingale if every variable X(t) is P-integrable, and if s ^ t, s, t e I, implies E[X(t) | T s ] < X(s), P-almost surely. An adapted process {X (t) : tel} with state space (R, 23) is called a sub-martingale if every variable X(t) is P-integrable, and if s ^ t, s, t e I, implies E[X(t) | T s ] 5= A(s), P- almost surely. If an adapted process is at the same time a super- and a sub¬ martingale, then it is called a martingale. The martingale in the following example is called a closed martingale. 1.18. Example. Let X ^ belong to L 1 (f2, £F, P), and let {£F t : te [0, oo)} be a filtration in T. Put X(t) = E [AT,*, | £F t ], t ^ 0. Then the process {X(t): t ^ 0} is a martingale with respect to the filtration {Tt : t e [0, oo)}. The following theorem shows that uniformly integrable martingales are closed martingales. 17 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites 1.19. Theorem (Doob’s theorem). Any uniformly integrable martingale {X(t):t> 0} in L 1 (fl, T, P) converges P -almost surely and in mean (i.e. in L 1 (fl, T, P),) to a stochastic variable X^ such that for every t ^ 0 the equality X(t) = E [A* | T t ] holds P -almost surely. Let F be a subset of L 1 (Q, T, P). Then F is uniformly integrable if for every e > 0 there exists a function g e L 1 (Q, T, P) such that t|/|>|p|} I/I ^ ^ £ f° r / e F. Since P is a finite positive measure we may assume that g is a (large) positive constant. 1.20. Theorem. Sub-martingales constitute a convex cone: (i) A positive linear combination of sub-martingales is again a sub-martin- gale; the space of sub-martingales forms a convex cone. (ii) A convex function of a sub-martingale is a sub-martingale. Not all martingales are closed, as is shown in the following example. 1.21. Example. Fix t > 0, and x, y e R d . Let {(fi, P*) ,(X(t),t>0),(ti t :t>0), (R n , T> n )} be Brownian motion starting at x e W ! , and put, as above, Pd(^x,y) = ^^exp The process s >—► p (t — s, X(s),y) is P x -martingale on the half-open interval [o,*)- 5.1. Stopping times. A stochastic variable T : —* [0, co] is called a stopping time with respect to the filtration {% : t ^ 0}, if for every t ^ 0 the event {T < t} belongs to 3y. If T is a stopping time, the process t is adapted to {3y : t ^ 0}. The meaning of a stopping is the following one. The moment T is the time that some phenomena happens. If at a given time t the information contained in £F t suffices to conciude whether or not this phenomena occurred before time t, then T is a stopping time. Let {(fi, P*), (X(t),t (R n , T> n )} be Brownian motion starting at x e R d , let p : R d —> (0, oo) be a strictly positive continuous function, and O an open subset of R d . The first exit time from O , or the first hitting time of the complement of O, defined by T = inf {t > 0 : X(t) e R d \0} is a (very) relevant stopping time. The time T is a so-called terminal stopping time: on the event {T > s} it satisfies s + T o d s = T. Other relevant stopping times are: r e = inf jt > 0 : J p(X(s))ds > fj , £ ^ 0. 18 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites Such stopping times are used for (stochastic) time change: r ? + o tf T£ = t£ + v , £, p > 0. Note that the mapping £ is the inverse of the mapping t h-> p (X(s)) ds. Also note the equality: {r^ < t} = |^p(A(s)) > £j, p(X(s)) ds > £ > 0. The mapping £ h-> 7 ^ is strictly increasing from the interval [0, p (X(s')) ds ) onto [ 0 ,co). Arbitrary stopping times T are often approximated by “discrete” stopping times: T = lim n ^oo T n , where T n = 2~ n [ 2 n T], Notice that T < T n+ 1 < T n ^T + 2~ n , and that {T n = k2~ n ) = {(k - l)2~ n < T < k2~ n }, ke N. 1.22. Theorem. Let (12, T, P) be a probability space, and let {U t : t ^ 0} be a filtration in £F. The following assertions hold true: (1) constant times are stopping times: for every t ^ 0 fixed the time T = t is a stopping time; ( 2 ) if S and T are stopping times, then so are min (A, T) and max (S', T); (3) IfT is a stopping time, then the collection TV defined by Tt = {A e T : A n {T < f] e for all t > 0} is a subfield of 3y (4) If S and T are stopping times, then S+Tods is a stopping time as well, provided the paths of the process are P -almost surely right-continuous and the same is true for the filtration [T t : t ^ 0 }. The filtration {T t : t ^ 0} is right-continuous ifT t = t ^ 0. The (sam¬ ple) paths t i—> X(t) are said to be P -almost surely right-continuous, provided for all t ^ 0 we have X(t) = lim s p A(s), P -almost surely. 19 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites The following theorem shows that in many cases fixed times can be replaced with stopping times. In particular this is true if we study (right-continuous) sub-martingales, super-martingales or martingales. 1.23. Theorem (Doob’s optional sampling theorem). Let (X(t) : t ^ 0) be a uniformly integrable process in L 1 (Q, £F, P) which is a sub-martingale with re¬ spect to the filtration (£F t : t ^ 0). Let S and T be stopping times such that S < T. Then E [X{T) \ T s ] > X(S), P -almost surely. Similar statements hold for super-martingales and martingales. Notice that X(T) stands for the stochastic variable u> > X ( T{pj )) (a;) = X (T(u),u). We conclude this introduction with a statement of the decomposition theorem of Doob-Meyer. A process {X(f) : t ^ 0} is of class (DL) if for every t > 0 the family {X(t) : 0 < r ^ t, t is an (9y) -stopping time} is uniformly integrable. An dy-martingale {M(t) : f A 0} is of class (DL), an increasing adapted process {A(t) : t ^ 0} in L 1 (f2,T, P) is of class (DL) and hence the sum { M(t ) + A(t) : t ^ 0} is of class (DL). If {X(f) : t ^ 0} is a submartingale and if p is a real number, then the process {max (X(t), p) : t ^ 0} is a sub-martingale of class (DL). Processes of class (DL) are important in the Doob-Meyer decomposition theorem. Let (f2,T, P) be a probability space, let {Si : t ^ 0} be a right-continuous filtration in J and let {X(t) : t ^ 0} be right continuous sub-martingale of class (DL) which possesses almost sure left limits. We mention the following version of the Doob-Meyer decomposition theorem. See Remark 3.54 as well. 1.24. Theorem. Let {AT(f) : t ^ 0} be a sub-martingale of class (DL) which has P almost surely left limits, and which is right-continuous. Then there ex¬ ists a unique predictable right continuous increasing process {A(t) : t ^ 0} with A(0) = 0 such that the process {X(t) — A(t ) : t ^ 0} is an T t -martingale. A process ( cu,t ) X(t)(co) = X ( t,u >) is predictable if it is measurable with respect to the u-field generated by {A x (a, b] : A e T a , a < b}. For more details on cadlag sub-martingales, see Theorem 3.77. The following proposition says that a non-negative right-continuous sub-martingale is of class (DL). 1.25. Proposition. Let (f2, T, P) be a probability space, let (Tt) tSs0 be a filtration of a-fields contained in T. Suppose that t > X(t) is a right-continuous sub¬ martingale relative to the filtration (3y) t>0 attaining its values in [0,oo). Then the family {X(t) : t ^ 0} is of class (DL). In fact it suffices to assume that there exists a real number m such that X (t) > —m P-almost surely. This follows from Proposition 1.25 by considering X{t) +rn instead of X(t). If t i—> M(t ) is a continuous martingale in L 2 (f2, £F, P), then t <—> |M(f)| 2 is a non-negative sub-martingale, and so it splits as the sum of a martingale t >—► 20 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites \M(t)\ 2 — (M, M) ( t ) and an increasing process t > (M < M) (t), the quadratic variation process of M(t). Proof of Proposition 1.25. Fix t > 0, and let t : Cl —> [0,£] be a stopping time. Let for m e N the stopping time r m : Cl —> [0, go] be defined by r m = inf {s > 0 : X(s) > m) if X(s) > m for some s < co, otherwise r m = co. Then the event (X(t) > m) is contained in the event {r m < r}. Hence, E [X(t) : X(t) > m] ^ E [X(t) : X(t) > m] < E [X(t) : r m < r] E [X(t) : r m < t] . (1-21) Since, P-almost surely, r m | co for m —*■ co, it follows that lim sup{E[X(r) : X(r) > m] : re [0, f] : r stopping time} = 0. m—> oo Consequently, the sub-martingale t >-*■ X(t) is of class (DL). The proof of Propo¬ sition 1.25 is complete now. □ It is perhaps useful to insert the following proposition. 1.26. Proposition. Processes of the form M(t) + A(t), with M(t ) a martingale and with A(t ) an increasing process in L 1 (0,T, P) are of class (DL). Proof. Let {X(t) = M(t) + A(t) : f 5= 0} be the decomposition of the sub-martingale {X(t) : t ^ 0} in a martingale [M(t) : t ^ 0} and an increasing process {A(t) : t ^ 0} with H(0) = 0 and 0 < r < t be any T r stopping time. Here t is some fixed time. For IVeNwe have E(|X(r)| : |X(r)| ^ N) < E(|M(r)| : |X(r)| ^ N) + E (A(t) : |X(r)| 5* N) <E(|M(f)| : \X(r)\ > N)+E{A(t) : \X(r)\ ^ N) <E(|M(*)| +A(t) : |X(r)| 5* N) < E ( \M(t)\ + A(t) : sup \X(s)\ > N Since, by the Doob’s maximality theorem 1.28, NF | sup \X(s)\ ^ N \ ^NF\ sup \M(s)\ ^—\+NFl sup A(s) > — f OsSssSt sS 2E (\M(t)\ + A(t )), it follows that 1 N N lim sup{E(|X(r)| : |X(r)| ^ N) : 0 < r < t, r stopping time} = 0. iV—► 00 This proves Proposition 1.26. □ First we formulate and prove Doob’s maximal inequality for time-discrete sub¬ martingales. In Theorem 1.27 the sequence i >—> X* is defined on a filtered probability space (Q, Tj.P) ieN , and in Theorem 1.28 the process t i—> X(t) is defined on a filtered probability space (Cl, T t .P) t>0 . 21 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites 1.27. Theorem (Doob’s maximal inequality). Let (Xj) ieN be a sub-martingale w.r.t. a filtration (3q) ieN . Let S n = max^^ X; be the running maximum of X, t . Then for any £ > 0, P [S„ < )e K], (1.22) where Xfi = X n v 0. In particular, if X* is a martingale and M n = max \Xf\, l ^ i^n then P [M n > £] < jE [\X n \ 1 {Mn>l} \ jE [\X n \\. (1.23) Proof. Let T£ = inf {i ^ 1 : X, ^ £}. Then P [S n ^ £\ = P \ji = i\. For each 1 < i ^ n, P[r ( - i] - E [l w „,l {T ,. j ,] ' Ie [X+Mr.-i )}. (1.24) Note that {re = i} e 3q, and Xf is a sub-martingale because X t itself is a sub-martingale while <p(x) = x + = x v 0 = max(x, 0) is an increasing convex function. Therefore \ p^n l{r^=i} | 5F»] = l { re = i}E \ X n | Tj] ^ 1 { r £= i } (lE [X n | 5)]) ^ lf T (= ijX i , and hence E [X+ lq T£= q] < E[X+l{ T£= q], Substituting this inequality into (1.24) and then summing over 1 < i < n then yields (1.22). The inequality in (1.23) follows by applying (1.22 to the sub-martingale \Xi\. □ A cate-Lucent www.alcatel-lucent.com/careers What if you could build your future and create the future? One generation’s transformation is the next’s status quo. In the near future, people may soon think it’s strange that devices ever had to be “plugged in.” To obtain that status, there needs to be “The Shift". 22 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites Next we formulate and prove Doob’s maximal inequality for continuous time sub-martingales. 1.28. Theorem (Doob’s maximal inequality). Let (X(t)) t>0 be a sub-martingale w.r.t. a filtration (T t ) t>0 . Let S(t ) = sup 0s£ss£4 X(s) be the running maximum of X{t). Suppose that the process t >—> X(t) is P -almost surely continuous from the right (and possesses left limits F-almost surely). Then for any i > 0, P[S(f) > e] « i|E[A'(i) + l (s(1| «,] « )e [A + (f)], (1.25) where X + (t ) = Xfi) v 0 = max(X(t),0). In particular, if t *—> X(t) is a martingale and M(t ) = sup |X(f)| ; then OsSssJt p [M(t)>e\ )e[|A(*)|1,=S )e[|A(()|], (1.26) Proof. Let, for every N e N, tjv be the (Tt) tSs0 -stopping time defined by t n = inf {t > 0 : X{t) + 5= N). In addition define the double sequence of processes X n ^(t) by X n , N (t ) = X (2~ n \2 n t] a t n ) . Theorem 1.28 follows from Theorem 1.27 by applying it the processes t h-> X n! N(t), n 6 N, N e N. As a consequence of Theorem 1.27 we see that Theorem 1.28 is true for the double sequence t >—> X ri ^r(t), because, essentially speaking, these processes are discrete-time processes with the property that the processes (n, t) >—>■ X nt N(t) + attain P-almost surely their values in the interval [0, N], Then we let n —> go to obtain Theorem 1.28 for the processes t >—► X (t a t/v), N e N. Finally we let N —*■ oo to obtain the full result in Theorem 1.28. □ 5.2. Additive processes. In this final section we introduce the notion of additive and multiplicative processes. Let E be a second countable locally compact Hausdorff space. In the non-time-homogeneous case we consider real¬ valued processes which depend on two time parameters: [ti,t 2 ) >-*■ Z (ti,t 2 ), 0 < t\ < <2 ^ T. It is assumed that for all 0 < t\ ^ <2 ^ T, the variable Z (ti,t 2 ) only depends, or is measurable with respect to, a (X(s) : t\ < s < t 2 ). Such a process is called additive if z (ti,t 2 ) = Z(ti,t) + Z (t,t 2 ), The process Z is called multiplicative if z (ti,t 2 ) = z (ti, t) ■ z (■ t , t 2 ), ti^t^ t 2 . Let p : [0, T] x E —> E be a continuous function, and let {X(t) : 0 < t < T) be an E- valued process which has left limits in E , and which is right-continuous (i.e. it is cadlag). Put Z (ti,t 2 ) = ^ p (s, X(s)) ds. Then the process (ti,t 2 ) i—>• Z (ti, t 2 ), 0 < ti < t 2 < T is additive, and the process (fi, t 2 ) i—> exp (Z (ti,t 2 )), 0 < ti < t 2 < T, is multiplicative. 23 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites Next we consider the particular case that we deal with time-homogeneous pro¬ cesses like Brownian motion: m Px), (M n , ® n )}, which represents Brownian motion starting at i e if An adapted process t Z(t) is called additive if Z (s + t) = Z (s) + Z (t ) o {) s , P x -almost surely, for all s, f P 0. It is called multiplicative provided Z (s + t) = Z (s) ■ Z ( t ) o $ s , P,, : -alinost surely, for all s, t P 0. Examples of additive processes are integrals of the form Z(t) = p (X(s)) ds, where x ^ p(x) is a continuous (or Borel) function on W ! , or stochastic integrals (Ito, Stratonovich integrals) of the form Z(t) = §yP (X(s)) dX(s). Such integrals have to be interpreted in some L 2 - sense. More details will be given in Section 6 . If t >—> Z(t) is an additive process, then its exponent t <—>■ exp (Z(t)) is a multiplicative process. If T is a terminal stopping time, then the process t l{r>t} is a multiplicative process. Let (X„) neN be a sequence of non-negative i.i.d. random variables each of which has density /i P 0 . Suppose that f n is the density of the distribution of 2 ” =1 Xj. Note “i.i.d.” means “independent, identically distributed”. Then P 1 fn(s)ds, Jo and hence rt n+1 n j f n+1 (s)ds = P = p Xj + x n+ i ^ t Jo J=1 j= i f = fn(p)fl(t- p)dp. It follows that rt r f rt ft fp fn(s)ds — f n+1 (p)dp= f n (s)ds- f n (s)f 1 (p-s)dsdp Jo Jo Jo Jo Jo f n (s)ds- fn(s ) fi(p — s)dpds Jo Jo Js fi(p-s)dp) ds rt rCC = fn(s) flip — s)dpd& Jo Jt rt rco = fn(s ) fi{p)dpds. JO Jt-S (1.27) If h(s) = Xe x ‘, then f„(s) \n g n -1 7 -rre _As . This follows by induction. (n — 1 )! 5.3. Continuous time discrete processes. Here we suppose that the process {( 0 ,T,P),(X(t): t> tp0),(5,S)} 24 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites is governed by a time-homogeneous or stationary transition probabilities: = P [*(*) = 3 I V(0) = i] = P [X(t + s)=j\ X(s) =i], i, j e 5, (1.28) for all 5^0. Here, S is a discrete state space, e.g. S — S — 7LJ 1 , — hi, or S = {0,7V}. The measurable space (fl,T) is called the sample or sample path space. Its elements uo e are called realizations. The mappings X(t) : —> 5 are called the state variables; the application t •—> X(i)(6j) is called a sample path or realization. The translation operators $*, t ^ 0, are mappings from to $2 with the property that: X(s) o $ t = X(s + i), P- almost surely. For the time being these operators will not be used; they are very convenient to express the Markov property in the time-homogeneous case. We assume that the Chapman-Kolmogorov conditions are satisfied: Pj,i (s + t) = i, je S, s,t> 0. (1.29) keS In fact the Markov property is a consequence of the Chapman-Kolmogorov identity (1.29). From the Chapman-Kolmogorov (1.29) the following important identity follows: P (s + t) = P(s)P(t), 5, t > 0. (1.30) The identity in (1.30) is called the semigroup property; the identity has to be interpreted as matrix multiplication. Suppose that the functions t ► p^(i), j, i e S, are right differentiable at t = 0. The latter means that the following limits exist: qj,i = lim • y ’ A|0 Pj,j ( A ) -Pj,i(0) A i, j e S. In the past four years we have drilled 81,000 km That's more than twice around the world. Whn am wp? fHSHHHH P We are the world's leading oilfield services company. Working 1 globally—often in remote and challenging locations—we invent, design, engineer, manufacture, apply, and maintain technology to help customers find and produce oil and gas safely. Who are we looking for? We offer countless opportunities in the following domains: ■ Engineering, Research, and Operations ^ ■ Geoscience and Petrotechnical ■ Commercial and Business A ^ If you are a self-motivated graduate looking for a dynamic career, apply to join our team. What will you be? careers.slb.com Schlumberger 25 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites We assume that Pj,i(0) = , where is the Dirac delta function: t = 0 if j # i, and 8 hJ = 1 . Put Q = (q ]a ) t je s- Then the matrix Q is a Kolmogorov matrix in the sense that q hl pz 0 for j ^ i and X? e s = 0 - If follows that Qi,i = ~ Xje 5 j^i Qjq T 0. The reason that the off-diagonal entries j ^ i, are non-negative is due to the fact that for j ^ i we have lim PjM) -PjM 40 t PjAt) 4 ( 0 ) 40 t lim 40 PjAt) t 0. In addition, we have E PS = y lim 40 PS PjAA ~ PjA°) lim y 40 4-* t — lim HjesPjA®) _ 1~1 _ q 4 o t 40 t (131) provided we may interchange the summation and the limit. Finally we have the following general fact. Let t >—> P(t ) be the matrix function t >—> (PjAA)i jeS' Then P(t) satisfies the Kolmogorov backward and forward differential equation: dP(t) dt = QP{t) = P(t)Q, t> 0. (1.32) The first equality in (1.32) is called the Kolmogorov forward equation, and the second one the Kolmogorov backward equation. The solution of this matrix¬ valued differential equation is given by P(t) = e tCi P( 0 ). But since P( 0 ) = (PjA®))jieS = AjAi jeS identity matrix, it follows that P(t) = e tC * . The equalities in (1.32) hold true, because by the semigroup property (1.30) we have: P (t + A(t)) - P (t) P(A(())»P(0) P(t)-P(t) A(t) A (t) Then we let A (t) tend to 0 in (1.33) to obtain (1.32). P(A(t))-P(0) m (1.33) 5.4. Poisson process. We begin with a formal definition. 1.29. Definition. A Poisson process m T, P), (N, X)} (see (1.46) below) is a continuous time process X(t), t ^ 0, with values in N = {0,1,...} which possesses the following properties: (a) For At > 0 sufficiently small the transition probabilities satisfy: Pi+i t i(At) = P [X (t + At) = i + 1 | X(t) = i\ = XAt + o (At ); p iti (At) = P [X (t + At) = i | X(t) = i] = 1 — XAt + o (At ); Pj,i(Xt) = P [X (t + At) = j | X(t) = i] = o (At ); Pj,i(At) = 0, j < i. (1.34) (b) The probability transitions (s, i\ t,j) >—► P [X (t) = j | X(s) = i], t > s, only depend on t — s and j — i. (c) The process (X(t) : t ^ 0} has the Markov property. 26 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites Item (b) says that the Poisson process is homogeneous in time and in space: (b) is implicitly used in (a). Note that a Poisson process is not continuous, because when it moves it makes a jump. Put Pi(t) = Pio(t) = p j+ i,j(t) = F[X(t) = j + i\ X(0) = i\, i, j e N. (1.35) 1.30. Proposition. Let the process m 3b P) ,(X(t),t>O),(0 t ,t>O), (N, N)} possess properties (a) and (b) in Definition 1.29. Then the following equality holds for all t ^ 0 and i e N: pm - < L36 > 1.31. Remark. It is noticed that the equalities in (1.42), (1.40), and (1.44) only depend on properties (a) and (b) in Definition 1.29. So that from (a), and (b) we obtain ^Piif) + A pfit) = iy e ~ Xt = APi-iW, * > 1, (1-37) and hence Pjfit) = ) = P [^(f) = j I -X'(O) = i] = jjz^y e ~ X \ 3 > *• ( 1 .38) If 0 ^ j < i, then Pjfit) = 0. Proof. By definition we see that Pj(0) = P [X(0) = j | X(0) = 0] = 5 0 j, and so pfiO) = 1 and pfi 0) = 0 for j / 0. Let us first prove that the functions t >—>■ pfit), i ^ 1, satisfy the differential equation in (1.45) below. First suppose that i ^ 2, and we consider: Pi (:t + At) - pfit) = P [X (t + At) = i] - pfit) i = J] P [X (t + At) = i, X(t) = k]~ pfit) k= 0 i = J] P [X (t + At) = * | X(t) = k] P [X(t) = k\- pfit) k= 0 = P [X (t + At) = i | X(t) = i] pfit) + P [X (t + At) = i | X(t) = i - l] Pi _fit) i—2 + 'Yj P [ x (t + At) = i | X(t) = k] pfit) - pfit) k= 0 i-2 = (1 - XAt + o (At)) pfit) + (XAt + o (At)) Pi-fit) + ^pfit)o(At) - pfit) k =o i = -XAtpfit) + XAtpi-fit) + ^pfit)o(At). (1.39) k =0 27 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites From (1.39) we obtain = -Api(t) + Xpi-i(t). (1.40) Next we consider i = 0: Po (t + At) - p 0 (t ) = P [X {t + At) = 0] - p 0 (t) = P [X (t + At) = 0 | X(t) = 0] P [X(t) = 0] - po(t) = P [X (t + At) = 0 | X(t) = 0]p 0 (t) — p 0 (t) = (—AAt + o(At))p 0 (t). (1.41) From (1.41) we get the equation ^Po(^) = — Apo(t). (1.42) For i = 1 we have: Px (t + At) -piit) = P[X(t + At) = 1] -pi{t) = P [X (t + At) = 1 | X(t) = 1] P [X(t) = 1] - p^t) + P [X {t + At) = 1 | X(t) = 0] P [X(t) = 0] = P [A it + At) = 1 | X{t) = 1] p^t) - p^t) + P [A (t + At) = 1 | X(t) = 0] poit) = (—AAt + o(At))pi(t) + (AAt + o(At))p 0 (t). (1-43) Join the best at the Maastricht University School of Business and Economics! gjpj* • 33 rd place Financial Times worldwide ranking: MSc International Business • 1 st place: MSc International Business • 1 st place: MSc Financial Economics • 2 nd place: MSc Management of Learning • 2 nd place: MSc Economics • 2 nd place: MSc Econometrics and Operations Research • 2 nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier 'Beste Studies' ranking 2012; Financial Times Global Masters in Management ranking 2012 Maastricht University is the best specialist university in the Netherlands (Elsevier) Master's Open Day: 22 February 2014 www.mastersopenday.nl | Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites From (1.43) we obtain: d dt pi(t) —Xpi(t) + Xpo(t). (1.44) By definition we see that Pj(0) = P [X(0) = j | X(0) = 0] = h 0 j, and so p o (0) = 1 and pj( 0) = 0 for j / 0. From (1.42) we get po(t) = e~ xt . From (1.40) and (1.44) we obtain 4 ( e M pi(t )) = Ae Ai pj_i(t), (xtY By induction it follows that pfit ) = — ^-e~ Xt . Proposition 1.30. i > 1. (1.45) This completes the proof of □ In the Proposition 1.33 below we show that a process m ^ P) > > o), (0 t , t> 0), (N, N)} (1.46) which satisfies (a) and (b) of Definition 1.29 is a time-homogeneous Markov process if and only if its increments are P-independent. First we prove a lemma, which is of independent interest. 1.32. Lemma. Let the functions pfit) he defined as in (1.35). Then the equality Pi (t)=V[X(s + t)-X(s) = i] (1.47) holds for all i e N and all s, t ^ 0. Proof. Using the space and time invariance properties of the process X(t) shows: 00 P [X(s + t)~ X(s) = i] = 2 P [X (s + t) - X(s) = i, X(s) = k ] k =0 oo = ^p[i(m) = * + fc, x(s) = k] k =0 oo = 2 P [X (s + t) = i + k | X(s) = k] P [X(s) = k] k =0 (space and time invariance properties of pfit)) 00 = = ( L48 ) k =0 The conclusion in Lemma 1.32 follows from (1.48). □ The following proposition says that a time and space-homogeneous process sat¬ isfying the equalities in (1.34) of Definition 1.29 is a Poisson process if and only if its increments are P-independent. 29 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites 1.33. Proposition. The process ]X(f) : t ^ 0} possessing properties (a) and (b) of Definition 1.29 possesses the Markov property if and only if its increments are ¥-independent. Moreover, the equalities P[A'(t) - X(s) -j-i]- P[A'(t) - j | A(a) = i] = p^(t -s)= (1.49) hold for all t ^ s 5= 0 and for all j ^ i, i, j e N. PROOF. First assume that the process in (1.46) has the Markov property. Let t n+ 1 > t n > ■ ■ ■ > ti > t 0 = 0, and let if., 1 ^ k ^ n + 1, be nonnegative integers. Then by induction we have P [X (D) - X (ti_ i) = ii, 1 ^ £ < n + 1] 00 = ^ P [X (tf) - X (te- 1 ) = i e , 1 n + 1, X (t n ) = k] k =o y P[X (U) - X (ti- 1 ) = 1 < l < n + 1, X (f n ) = fc] P [X (ti) - X (ti- 1 ) = ii, 1 n, X (t n ) = k ] x P [X (tf) - X (ti- 1 ) = ii, 1 < t < n, X (t n ) = fc] 00 = J] P [X (t n+ i) - X (t n ) = i n+ 1 | X (^) - X (ti- 1) = **, X (t n ) = fc, k=0 1 < £ < n] x P [X (t*) - X (ti- 1 ) = i e , X (t n ) = fc, 1 < £ < n, ] (Markov property) 00 = 2 P [X (t n+1 ) - X (t n ) = i n+1 | X (t n ) = fc] k =0 x P [X (ti) - X (ti- 1 ) = ii, 1 i n, X (t n ) = fc] 00 = J] P [X (t n+ i) = i n+ 1 + fc | X (f n ) = fc] k—0 X P [X (b) - X (t^_i) = ii, 1 < £ < n, X (t n ) = fc] (homogeneity in space and time of the function t >—> Pi n+1 (t)) 00 = 2 Pi n+1 (tn+1 - t n ) P [X (ti) ~ X (ti- 1 ) = ii, 1 ^ n, X (t n ) = fc] k =0 (apply equality (1.47) in Lemma 1.29) 00 = J>[X (t n+1 ) - X (t n ) = i n+1 ] k—0 P [X (tf) - X (ti-i) = ii, 1 < t < n, X (t n ) = fc] 30 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites = P [x (t n+ 1 ) - X (t n ) = i n+ 1 ] P [X (tg) - X (tg- 1 ) = t*, 1 < i < n]. (1.50) By induction and employing (1.50) it follows that n P [X (tg) - X 0 tg _i) = i* 1 < £ < n] = f] P [X (tg) - X ( tg _,) = i<] €=1 n = \\pi IL {tg-tg- l ). (1.51) £=i We still have to prove the converse statement, i.e. to prove that if the increments of the process X(t) are P-independent, then the process X(t) has the Markov property. Therefore we take states 0 = t 0 , ti,... ,i n , t„+ i, and times 0 = to < tg < • ■ ■ < t n < t n+ 1 , and we consider the conditional probability: P [X (tn+i) = i n +1 | X (to) = to, • • •, x (t n ) = i n ] _ P [X (t n+ i) = i n+ 1, X (t 0 ) = t 0 ,...,X (t n ) = t n ] P [X (t 0 ) = t 0 ,..., X (t n ) = i n ] _ P [X (t 0 ) = t 0 , X (tg) - X (tg- 1 ) =ig- ig-i, 1 ^ l ^ n + 1] P [X (t 0 ) = t 0 , X (tg) - X (tg- 1 ) = t f - t^_i, 1 ^ £ < n] (increments are P-independent) = P [X (t n+ 1) - X (t n ) = i n+ 1 - t n ] = P [X (t n +i) = i n +i I x (t n ) = i n ] . (1.52) > Apply now REDEFINE YOUR FUTURE AXA GLOBAL GRADUATE PROGRAM 2015 redefining /standards £ 31 Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites The final equality in (1.52) follows by invoking another application of the fact that increments are P-independent. More precisely, since X (t n + 1 ) — X (t n ) and X (t n ) — X(0) are P-independent we have P [X (t n+ 1) = i n+ 1 | X (t n ) = i n \ = P [X (t n+ 1) - X (tn) = in+l - in, X (t n ) - X (0) = Z n ] P[X(t n )-X(0)=i n ] = P [X (t n + 1) - X (t n ) = i n+ 1 - i n ] . (1.53) The equalities in (1.49) follow from equality (1.47) in Lemma 1.32, from (1.53), from the definition of the function Pi{t) (see equality (1.37)), and from the explicit value of pAt) (see (1.36) in Proposition 1.30). This completes the proof of Proposition 1.33. □ Let (fl, T. P) be a probability space and let the process t h-> N(t) and the proba¬ bility measures P j, j e N in j(fi, T, P 'j)j eN , (N(t) : t ^ 0), (d s : s ^ 0), (N, N) j have the following properties: (a) It has independent increments: N(t + h) — N(t ) is independent of = ex (N(s) - X(0) : 0 < s ^ t). (b) Constant intensity: the chance of arrival in any interval of length h is the same: P [N(t + h)~ N(t) >1] = Xh + o(h). (c) Rarity of jumps ^ 2: P [N(t + h)~ N(t ) >2]= o(h). (d) the measures P 7 , j ^ 1, are defined by: Pj[X] = P [X | N( 0) = j]; moreover, it is assumed that Pq[X(0) = 0] = 1. The following theorem and its proof are taken from Stirzaker [126] Theorem (13) page 74. 1.34. Theorem. Suppose that the process N(t ) and the probability measures satisfy (a), (b), (c) and (d). Then the process N(t) is a Poisson process and V j [N(t) = k]= ( ^Xye~ x ^\ k>j. (1.54) PROOF. In view of Proposition 1.33 it suffices to prove the identity in (1.54). To this end we put fn(t) = Po [N(t) =n]= P [N(t) = n | X(0) = 0] = P [N(t) - N( 0) = n ]. Then we have, for n ^ 2 fixed, n f n (t + h) = P 0 [X (t + h) = n] = J] P 0 [X (/ I //) N(t) = k, N(t ) = n - k] k =0 32 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Stochastic processes: prerequisites (the variables N(t + h) — N(t ) and N(t) are P 0 -independent) n = 2 P 0 [N (t + h) - N(t) = k] x P 0 [N(t) = n - k] k =0 = P 0 [N (;t + h ) - N(t) = 0] X P 0 [N(t) = n ] + P 0 [AT (t + h) - N(t ) = 1] x P 0 [iV(t) = n - 1] n + 2 Po [N (t + h) — 7V(t) = fc] X P 0 [N(t) = n - k] k =2 = (1 - P 0 [iV (t + h) - N(t ) ^ 1]) X P 0 [iV(i) = n] + P 0 [AT (t + h)- N(t) > 1] x P 0 [iV(t) = n - 1] - P 0 [iV (t + h) - N(t ) ^ 2] x P 0 [AT(t) = n - 1] n + Y l Wo[N(t + h)- N(t ) = fc] x P 0 [iV(t) = n - /c] k=2 = (1- Xh + o(h )) x / n (t) + (Ah + o(h)) f n -i(t) + o(h ) /»—*(t) k =1 = (1 - XH) f n (t ) + Xhf n -i(t) + o(h). (1.55) Observe that a similar argument yields fi(t + h) = (1 - Xh ) /i (t) + Xhf 0 (t) + o(h), (1.56) and also fo(t + h) = (1 — Ah) fo(t) + o(h). (1.57) From (1.55), (1.56) and (1.57) we obtain by rearranging, dividing by h and allowing h [ 0: /,',(*) = -A/n(<) + A/„_i(t), »>1, /;<<) - -a/m. These equations can be solved by induction relative to n. A alternative way is to consider the generating function 00 00 G(s,t) := E 0 [e sJV W] = £ s"P 0 [iV(t) = n] = £ s n /nW- n=0 Then ~^r ~ = M s ~ l)G(s,t), and so G(s,t ) = e A ^ s_1 ). It follows that (Xt) n Po \N(t) = n] = f n (t) = e~ xt ——. Consequently, for k ^ j we obtain n\ Fj [N(t) = k]= P [AT(t) = k | iV(0) = j] = P [N(t) - N(0) = k-j \ N(0) = j] = P [N(t) - N(0) = k - j] = e ~ A(fc ~ j) nf- j)\ = ( RHS ° f ^ L54 )- This completes the proof of Theorem 1.34. □ 33 Download free eBooks at bookboon.com Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains CHAPTER 2 Renewal theory and Markov chains Our main topic in this chapter is a discussion on renewal theory, classification properties of irreducible Markov chains, and a discussion on invariant measures. Its contents is mainly taken from Stirzaker [126]. 1. Renewal theory Let (X r ) re ^ be a sequence of independent identically distributed random vari¬ ables with the property that P[X r > 0] > 0. Put S n = 2r=i X r , <5o = 0, and define the renewal process N(t) by N(t) = max{n : S n < t}, t ^ 0. The mean m(t) = E [7V(t)] is called the renewal function. We have N(t) ^ n if and only if S n A t, and hence P [N(t) = n] = P [S n ^ t] — P [5 n+ i ^ t], and (2.1) 00 00 E [N(t)] = 2 P [N(t) > r] = 2 P [S r < t ]. (2.2) r— 1 r— 1 For more details see e.g. [4] (for birth-death processes) and [126] (for renewal theory). 2.1. Theorem. IfE[X r \ > 0, then N(t ) has finite moments for all t < go. Proof. Since E [X r ] > 0 there exists e > 0 such that P [X r ^ e] ^ e. Put M(t) = max [n : sYT r =i 1 {x r ^e} < t}. Since ejfr=i 1 {x r >e} < lTr=i X r it follows that N(t) ^ M(t), and hence, with m = 00 00 E [N(t)] A E [M(t)] = F [M(t) > n] = P n=l n=l oo = 2 2 P [Xj > £, j £ A, Xj < s, j f A] n= 1 Ac{l,...,n} ; oo = 2 2 P[Xi> e]* A (1 - P [X x ^ e]) n ~* A n= 1 Ac{l,...,n} ; oo nAm / \ = E E (o p [A »T (i - p pa == n=lfc=0 m oo / \ S " (l-P[XiS£])-‘ k =0 ' n e 2 ^ t r=l 35 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2 «• ixi > T £ fc =0 1 P [X r > n=0 t S n + k n (1-PpC >e]) r + 1 (2.3) In the final equality in (2.3) we used the equality: ^ 71 = 0 n + k n z k = a for \z\ < 1. The inequality in (2.3) shows Theorem 2.1. { k +1 □ It follows that E [X(t)] is finite whenever E [X r ] is strictly positive. This fact will be used in Theorem 2.2. 2.2. Theorem. The following equality is valid: E[S m+1 ]=E[X 1 ]E[N(t) + l]. The equality in Theorem 2.2 is called Wald’s equation. Proof. The time N(t) + 1 is a stopping time with respect to the filtration T n = a (X r : 0 < r < n) = a (S r — rE [Xi] : 0 < r < n). Notice that the process n >—>■ S n — nE [Xi] is a martingale, and hence E [5(jv( t ) + i) An - ((N(t) + 1) a n) E [XJ] = E [S m)+ i ) a0 - {(N(t) + 1) a 0) E [X,]] = 0. (2.4) Since E [X(t)] is finite, from (2.4) we get by letting n tend to go: 0 = lim E [S(tf(t)+i)An - ((N(t) + 1) a n) E [X x ]] = E [S (w(t|+1) - (( N(t ) + 1)) E [A',]]. (2.5) Consequently, the conclusion in Theorem 2.2 follows. □ 2.3. Theorem. Let (X r ) reN be a sequence of independent, identically distributed random variables such that P [X r = 0] = 0. Put So = 0 and S n = X!r=i ^r- Let the process N(t ) be defined as in (2.2). Let Fit ) be the distribution function of the variable X r . Put m(t) = E [X(t)]. Then m(t ) satisfies the renewal equation: ft °° m{t) = Fit) + m(t - s)dF(s) = ^ (/j,* F ) k [0,t], (2.6) k =1 where fj,F{a,b] = F(b) — F(a), and pti * b] = J* l( a ,6](s + t)dfj,i(s)d/j, 2 (t), 0 < a < b (i.e. convolution product of the measures /ii and fi 2 )- Moreover, ( rCO \ r*00 rCO 1 - J e~ Xs dF(s)J x A J e~ xt m{t) dt = J e“ As dF(s). If X r are independent exponentially distributed random variables, and thus the process (N{t) : t ^ 0) is Poisson of parameter A > 0, then m{t) = A t. 36 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains PROOF. On the event {Xi > t} we have N(t) = 0, and hence by using conditional expectation we see m(t) = E[N(t)] = E[N(t)l {Xl<t} ] E [E [iV(t)l {Xi (Vi)]] - E [l {Jfl «,E PW — JV (ATi) | <7- (X0]] + E [l |Xl «,E [N (A',) | o (A t )]] (on the event {X\ < t) we have N (Xi) = 1) - E [1( X ,«)E PW (*i)]] + E [l{X lS £t}E [iRW]] (the distribution of N(t)—N(s ), t > s, is the same as the distribution of N(t—s )) = E [l { x^ t} E [N (t -X x )\a (X:)]] + E [1 {Xl & } E [1 | o (X,)]] = E [N (t — Xi) l{Xi^t}] + E [l{Xi<t}] = I m(t — x) dF(x) + F(t). This completes the proof of Theorem 2.3. (2.7) □ Empowering People. Improving Business. Norwegian Business School is one of Europe's largest business schools welcoming more than 20,000 students. Our programmes provide a stimulating and multi-cultural learning environment with an international outlook ultimately providing students with professional skills to meet the increasing needs of businesses. B! offers four different two-yea i; full-time Master of Science (MSc) programmes that are taught entirely in English and have been designed to provide professional skills to meet the increasing need of businesses.The MSc programmes provide a stimulating and multi¬ cultural learning environment to give you the best platform to launch into your career * MSc in Business * MSc in Financial Economics * MSc in Strategic Marketing Management NORWEGIAN BUSINESS SCHOOL EFMD EQUIS *ffi * MSc in Leadership and Organisational Psychology www.bi.edu/master Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.4. Lemma. Suppose P[X r < oo] = 1. lim N(t) = go, t —>00 Then P -almost surely. ( 2 . 8 ) Proof. Put Z = lim N(t) = supiV(t). Observe that t_>0 ° o hence by letting t —> go, the event {Z < go} is contained in thus N(t )+1 2 A * fc = l Z+l u w - r=l > t, go}, and and U w - »} r —1 < 2 P [X r = oo] = 0. r— 1 The result in Lemma 2.4 follows from (2.9). [Z < go] = P z+i |^J {X r = oo} , Z < GO r— 1 < p (2.9) □ Since lim^, x , N(t) = go P-almost surely, we have lim ~ = 1 P-almost surely. The following proposition follows from the strong “law” of large numbers (SSLN). 2.5. Proposition. Let (X r ) ? , eN be a sequence of non-negative independent, iden¬ tically distributed random variables in L 1 (f2,£F, P) such that P [X r < go] = 1. Then lim = lim ' S " (,) t->CO N(t) + 1 t->oo N(t) = E [Xj , P -almost surely. (2.10) 2.6. Theorem (First renewal theorem). Let the hypotheses be as in Proposition 2.5. Then ^ P -almost surely. (2-11) lim t->GO t EM’ PROOF. By definition we have S)v(t) ^ t < <S)v(t)+i, therefore 5 N(t) < t < X(f) + 1 <Sjv(p+i N(t)^N(t)' s N(t) N(t) + 1 (2 ‘ 12) The result in (2.11) now follows from (2.12) in conjunction with (2.8) and (2.10). This proves Theorem 2.6. □ The proof of the following theorem is somewhat more intricate. 2.7. Theorem (Elementary renewal theorem). Let the hypotheses be as in Proposition 2.5. Ts above, put m(t ) = E[iV(f)]. Then , m(t) 1 lim - = — y—TT. t-> oo t E [Xi] (2.13) 38 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.8. Remark. From Theorem 2.6 and 2.7 it follows that the family is uniformly integrable. Here we use Scheffe’s theorem. Proof of Theorem 2.7. This equality has to be considered as two in¬ equalities. First we have t < 5jv(t)+i, and hence by Theorem 2.6 we see t < E [S m+1 ] = E [X,] (E [N(t)] + 1) = E [X,] (m(t) + 1). (2.14) The inequality in (2.14) is equivalent to m(t ) ^ 1 1 ~T ^ E[Xi] ~~ T (2.15) From (2.15) we see lim inf £—>oo m(t) t ^ lim inf £—> oo 1 E [Xi] (2.16) For the second inequality we proceed as follows. Fix a strictly positive real number a, and put N a (t) = max{n e N : 2r=i m i n ( a >^) ^ ^}- Then N(t) < N a (t). Moreover, by Theorem 2.2 we have t > E [<Sjv a (t)] = E [<Sjv a (p+i - min (a, Xjv a (f)+i)] = E [min (a, X x )] E [N a (t) + 1] - E [min (a, X^q+i)] ^ E [min (a, Xi)] E [N(t) + 1] — a = ( m(t ) + 1) E [min (a, Xi)] — a. (2.17) Hence, from (2.17) we obtain: m(t) 1 a — E [min (a, Xi)] t E [min (a, Xi)] + tE [min (a, Xi)] From (2.18) we deduce: lim sup £—>oo m(t) t < 1 E [min (a, Xi)] ’ By letting a —* go in (2.19) we see for all large a > 0. lim sup £—>oo m{t) t 1 E [X^ ‘ (2.18) (2.19) ( 2 . 20 ) A combination of the inequalities (2.16) and (2.20) yields the result in Theorem 2.7. □ Next we extend these renewal theorems a little bit, by introducing a renewal- reward process (R n ) neli , where “costs” are considered as negative rewards. We are also interested in the cumulative reward up to time t: C(t ) (the reward is collected at the end of any interval); C\ (t) (the reward is collected at the start of 39 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains any interval); C P {t ) (the reward accrues during any given time interval). More precisely we have: N(t) c(t) = y Rj , terminal reward at the end of time interval, (2.21) i =! N(t)+1 Ci{t) = ^ Rj, initial reward at the beginning of time interval, (2.22) j = 1 N(t) Cp(t ) = ^ Rj + P N (t)+ 1 , partial rewards during time interval. (2.23) 3 = 1 For the corresponding reward functions we write c(t) = E [C(t)], d{t) = E [Ci(t)] and c p {t) = E [C P (t )]. (2.24) , r , C(t) CAt ) , C P (t) T . We are interested m the rates of reward: — : —, — : —, and — : —. It is assumed t t t that the renewal process N(t) is defined by inter-arrival times X r , reN. As above these inter-arrival times are non-negative, independent and identically distributed on a probability space (fl, IF, P). It is also assumed that the renewal- reward process R n , n e N, consists of independent and identically distributed random variables in the space L 1 (f2, £F, P). 40 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains The following theorem will be proved. 2.9. Theorem (Renewal-reward theorem). Suppose that 0 < E [XJ < go, eo^i] < go, and that the sequence (n 1 T > n ) neN is uniformly bounded in n e N and u>, and has the property that lim n ^oo n~ l P n = 0, P-almost surely. Let the notation be as in (2.21), (2.22), (2.23), and (2.24). Then the following time E [i?i] average limits exist P-almost surely and they are identified as iim m = , im m = Um m t — >00 t t—► 00 t ► oo t The following equalities hold as well: lim ^ = hrn ^ = lim ► 00 t t—> oo t t—► 00 E[i?i] E [Xi] ’ cp(t ) E[X x ] P-a/most surely. _ E[R X ] = E [Xi]' (2.25) (2.26) Observe that the quotient E[i?i] can be interpreted as the “expected reward E M accruing in a cycle” divided by “expected duration of a cycle”. Other conditions on the sequence (P n : n e N) can be given while retaining the conclusion in Theorem 2.9. For example the following conditions could be im¬ posed. The sequence (P n : n e N) is P-independent and identically distributed, or there are finite deterministic constants c\ and C 2 such that \P n \ ^ Cin + C 2 \R n \ P n [Pn and lim — = 0. In these cases the sequence ( — n—fco n \n Pn grable and lim — = 0 P-almost surely. n—>oo n Proof. By employing Theorem 2.2 and the strong law of large numbers we have : n e N ) is uniformly inte- lim C -21 = Urn _ EM, t >co t t—*co N(t) t E [Xl] In exactly the same manner, with N(t) + 1 replacing N(t). we see lim t >GO t E [i?i] P n — 7 —-r. By hypothesis we know that lim — = 0 P-almost surely. Since E [AiJ n— >00 n lim N(t) = go P-almost surely (2.27) Cfit) 00 we see that lim ..... AT ) +1 _ g_ 'ppjg together with (2.27) shows that t -00 N(t) + 1 & v ; lim t—* 00 Cp(t) E[i2i] t E [Xi]' These arguments take care of the P-almost sure convergence. Next we consider the convergence of the time averaged expected values. For convergence of time average of the reward function cfit) = E[Q(£)] we use 41 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains Wald’s equation (see Theorem 2.2) and the elementary renewal Theorem 2.7. More precisely we have: c i (t)=E[C i (t)]=E N(t) +1 2 * _ 3 = 1 = E [Ri] (E [N(t)] + 1). (2.28) Then we divide by t, take the limit in (2.28) as t tends to oo. An appeal to Theorem 2.6 then shows the existence of the limit lim — ^ = — [—4 which co t E[W] Rn is the second part of (2.26) in Theorem 2.9. First observe that lim —- = 0 n—*co n P-almost surely. This can be seen by an appeal to the Borel-Cantelli lemma. In fact we have T n —1 \Rn\ - > £ n 00 = 2> 71=1 \\Ri\ ^ ] -> n £ rcc < P Jo \\Ri\ I - > X £ dx < E 1-1 (2.29) Rn From (2.29) together with the Borel-Cantelli lemma it follows that lim —- = 0 n— > co n P-almost surely. Consequently, the sequence -< — : n e N > is P-uniformly in¬ tegrate. Then we have I R n »«)+i| „ N(t)+ l2S )+1 |ii t and hence by Wald’s equality t N(t) + 1 ’ E R W(t)+l| t ^ E N(t) + 1 Zk=l +1 I Rr t N(t) + 1 m(t) + 1 t (2.30) E[|i?!|], (2.31) By the strong law of large numbers and by the elementary renewal theorem 2.7 we see that the families of random variables s£? +1 \k i _ N ( t )+1 stir 1 \k N(t )+1 is uniformly integrable. Consequently the family N(t) + 1 ’ f R N (t )+1 l t t > 0, (2.32) : t > 0 > is uniformly integrable, and hence it converges pointwise and in L 1 (12, T. P) to 0. Since c(t) 1 E 'N(t)+1 S ft fc=1 E [Ri] (2.33) This proves the first part of The right-hand side of (2.33) converges to . E [Xi\ (2.26) in Theorem 2.9. In order to prove the third part we need the uniform integrability of the family Pi N(t) +1 t : t ^ 1 >. This fact is not entirely trivial. 42 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains Let the finite constant C be such that \P n +i\ < C(n + 1) for all n e N and P- almost surely; by hypothesis such a constant exists. From Remark 2.8 it follows that the family N(t) + 1 is uniformly integrable. Since N(t) + l\ t it follows that the family PN(t)+ 1| N( t) + 1 < + 1 N(t) + 1 Pnw+i (2.34) t : t ^ 11 is uniformly integrable as well. If ' ’ in L 1 (f^fF, P) as well as P- t t co then N(t) t co, and lim 1 ’ w 1 ’ t->oo t E [Xi] almost surely: see Lemma 2.4, theorems 2.6, 2.7, and Remark 2.8. From (2.34) E [|P/v (t)+1 |] it follows that lim ——-— = 0, which concludes the proof of Theorem t->0 0 t 2.9. □ Brain power By 2020, wind could provide one-tenth of our planet’s electricity needs. Already today, SKF’s innovative know¬ how is crucial to running a large proportion of the world’s wind turbines. Up to 25 7o of the generating costs relate to mainte¬ nance. These can be reduced dramatically thanks to our (^sterns for on-line condition monitoring and automatic lul|kation. We help make it more economical to create cleanSkdneaper energy out of thin air. By sh?fe|ig our experience, expertise, and creativity, industries can boost performance beyond expectations. Therefore we need the best employees who can kneet this challenge! Power of Knowledge Engineering Plug into The Power of Knowledge Engineering. Visit us at www.skf.com/knowledge 43 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 1.1. Renewal theory and Markov chains. Next we consider this re¬ newal theory in the context of strong Markov chains. Let (Q, T, P) be a proba¬ bility space and let X m , me N, be a Markov chain on (Q, £F, P) with state space (S, S). Fix two states j and k e S. Define the sequence of stopping times Tjf\ reN, as follows: t: (r+l) = min {n > : X„ = k '}■ T fc (0) = 0. (2.35) If X n 4= k for n > Tjf\ then we put Tff^ L> = go. The sequence of differences — T^~ l \ r A 1, are Pj-independent and identically distributed. 2.10. Theorem. Let f : [0,go] x S —» E be a bounded measurable function. Then -i(r+l) T! (r+s) = Ti r) + rr o o w /pNb !«'>-, IVJ ->( r ) on | < coj, and (2.36) = E,- = E,- / t; -,(r+s) AT ,(r + s)) 1{ T W <00 J | (X [T^\X n(r) f (Tj: r+s) , X T( r + s )) Imd = cj i—> E 'X (cd) ^ ) M {TW<oo} 60/ cr (T, i(r) y - / (TV) + T* (V) , v r ,, V) (V))J i {t ,. )<oo} h y - / (TV) + T 1 (V), v T ,. V) (V))] i {T (.. <<o( (u.). (2.37) Consequently, conditioned on the event j J) , :i < col /fte stochastic variable = cu Et 7^( r + s ) _ t 1 1 fc 1 k 00 and the a-field T„m are Pj -independent. Suppose that P& if < oo = 1 and P,- T fc (1) < oo > 0. TTien E,- T fc (r+1) < oo T fc (1) < oo and the variables T ( f ' +l} — T^\ reN, have the same distribution with respect to the probability measure A h-> P^ A | T fc (1) < oo Here Pj(H) = P [H | X 0 = j], A e 3 r , j e S. Theorem 2.10 is a consequence of the strong Markov property. PROOF. First we prove (2.36). On the event |t)^ < ooj we have T^ r+1 ^ = min j n > : X n = A; j = min j n > : X n _ T ( r ) o $ T w = /c j = T fc (r) + min jn - T fc (r) ^ 1 : ^ n _ T U) ° $ T w = fc J = T fc (r) + min | m ^ 1 : X m o = fc j 44 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains - + T k ( 1 ) ° &r(r) ■ (2.38) The equality in (2.38) shows (2.36) in case s = 1. We use (2.38) with s respec¬ tively r + s instead of r to obtain (2.36) by induction on s. More precisely we have rf + T k (*+1) o v> = rf + ( T i” + n x ' ° >U> I ° d 3 ) l(l) = Ti r) + Tl s) o riUo + o d T (, )+r ( S ) otf k ^(1) .O’*) (2.39) (induction hypothesis) h r+ ' ) +if o tf T(r+ „ i(r+s+l) (2.40) where in (2.39) we employed (2.38) with s instead of r and in (2.40) we used r + s instead of r. The equality in (2.40) shows (2.36) for s + 1 assuming that it is true for s. Since by (2.38) the equality in (2.36) is true for s = 1, induction shows the equality in (2.36). Next we will prove the equality in (2.37). From equality (2.36) we get % / [T t -r(r+s) x„ ,(r + s) ) 1 { T i rK ’} 37 ,(r) — E j Ej /I if + r;"ot H ,x W k ^ L 'j’( r ) i (r) j X {TW<oo} 37 (r) / (r)+ (s)o ^ x „ ) o} I (the variable is 37,(r) -measurable in combination with the strong Markov 1 k property) = oj = UJ E X ( r \ (i d) T k (“) / ^rf(w) + T' t ”,X T t„ JJ 1j t w <co) (ui) n(s) Eb /( i rfM + rr,A' T <., /l ji {T ,. )<oor The equalities in (2.41) show that the first, penultimate and ultimate quantity in (2.37) are equal. Another appeal to the strong Markov property shows that the first and second quantity in (2.37) coincide. Since on the event j< ooj the equality X T ( r ) = k holds, the second and third quantity in (2.37) are equal as well. This proves that all quantities in (2.37) in Theorem 2.10 are the same. We still have to prove that on the event j T'j?' 1 < ooj the stochastic variable T A ( r+ ' s) — T^ ] and the a-field 3 r m are P ; -independent. This can be achieved k as follows. Let the event A be $ t (t )-measurable and let g : [0, oo] —► M be a k bounded measurable function. Then we have -l(s) .H- (2.41) E,- a (if * 1 - if 1a 1 {tM<oo}J _ E i g ( T^ o tf T ( r) J 1 A 1 j T w <00 j (2.42) — E j Ej g (t^ s) o 1 A | 1| t M <00 | 45 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains = E,- E,- 9 T, <s) - < 9 . 1 k k 1a 1 {TW<co} (strong Markov property: (2.37)) — Ej = Et E x (r) 9 T h W _ lAl K r) <co} — Ej E i 9 T h n(s) 1 A 1 /rp(r-) {T«<cc}_ 9(rf)] Ej 1a1| t w < oo | (another appeal to the strong Markov property: (2.37)) (2.43) Ej = E,- Ej E,- g(T^ s) ) o^ M l « A {rW<®} I - Ej p A l| r w <oo ^ P,- g (V£ ) o # t m) l| T w <00 | | E,- T fc (r) < oo 1 A I 7L (r) < oo (use (2.36)) = E,- Ej E,- 9 (h’ +,> - T k ] ) l { Tj><® } I J rj>JJ E j L A t, w < oo 0 P 1 / ->(»•+*) _ fc ) 1 {tM<oo}J [ 1a I T k ri r) < oo (2.44) What do you want to do? No matter what you want out of your future career, ar employer with a broad range of operations in a load of countries will always e the ticket. Working within the Volvo Group means more than 100,000 friends and agues in mofe than 185 countries all over the world. We offer graduates great career opportunities - check out the Career section at our web site www.voivogroup.oom. look forward to getting to know you! VOLVO AB Volvo ipublj wvrtv.vc*WBP&jp.:«T. Vouro Trucks I Renault Trucks I Hack Trucks I Volvo Buses I Volvo Constbuctioh Equipment I Volvo Pekta I Volvo Aero I Voivo IT Voivo Financial Services I Volvo 3P I Volvo Powertrain I Volvo Pacts I Vouro Technocogv | Venire Loots tics I Business Area Asia Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains From (2.44) the Pj-independence of T^ r+ '^— and the cr-field jF' ( r ) conditioned k on the event j < ooj follows. Since the expressions in (2.42) and (2.43) are equal it follows that the P j (r) T> does not depend on r, provided that T fc (1) < oo -distribution of the variable T, (r+l) in < co}\{Ti r) < co} By the strong Markov property it follows that P; t: (r+l) < GO = P j T k ] < 00 Pfc = 0. T k ] < 00 (2.45) (2.46) Since, by assumption, T fc (1) < oo = 1, (2.46) implies that the probabilities P; Tl r) < oo do not depend on r e N, and hence (2.45) follows. This proves □ Theorem 2.10. 2.11. Definition. Let j e S. If Pj (or persistent). If Pj T (1) < oo ri 1} < oo recurrent state for which E , 3 (1) 1, then j is called recurrent < 1, then j is called a transient state. A = go is called a null state. A recurrent state for which Ej (i) < oo is called a non-null or positive state. From (2.46) it follows that Pj rj r) < oo = ^ Tj 1} < oo ted to be visited infinitely , and hence if a N k = Xm=i 1{ x n =k } be the number of visits to the state k , and put u k = Ej [N k ] = E [N k | A 0 = j]. Then < 4 - 2 = 2 P [W„ = * J Wo = jr] - n —1 n —1 We also have {N k ^ r + l} = jr fc (r+1) < ooj and hence by (2.46) we get P j [N k ^r + l]= P j [N k > 0] P fc [N k > Of (2.47) P, T k +l) < 00 = P, T fc (1) < oo P k T fc (1) < oo (2.48) From (2.47) and (2.48) it follows that ^ = Yi Pik = 2 P [X = k I Xo = j] = Fj [N k > 0] 2 P fc [N k > 0] r . n—1 n —1 r—1 (2.49) Suppose that the state j communicates with k , i.e. suppose that > 0 for some integer n ^ 1. From (2.49) it follows that the state k is recurrent if and only if XIn=i Pjk = 00 • Th e state k is transient if and only if Yju=\ P% oo (n) < 00 . 47 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.12. Theorem. Suppose that the states j and k intercommunicate. Then either both states are recurrent or both states are transient. PROOF. Since the states j and k intercommunicate the exist positive inte¬ gers m and n such that p^ > 0 and pj”' ) > 0. For any positive integer r we then have (m+r+n) (m) Jr) Jn) Pjj ^ Pjk PkkPkj too (r) (2.50) By summing over r in (2.50) we see that pfj < 00 if and only if X!^=i Pkl < oo. From this fact together with (2.49) the statement in Theorem 2.12 follows. □ 2.13. Definition. A Markov chain with state space S is called irreducible of all states communicate, i.e. for every j, k e S there exists neN such that p^l > 0. If X is irreducible and all states and one, and so all states, are recurrent, then X is called recurrent. 2.14. Theorem. Let X be a recurrent and irreducible Markov chain. Put (2.51) Then 0 < v* < go, j, k e S, and Vj = X \ieS v iPij■ ^ n °^ er words the vector (vj : j e S) is an invariant measure for X. Proof. First we prove that 0 < v 1 - < go. Therefore we notice that r T 1 ( f ) 1 k ± k II 2 1 1 x u = i ) \ x o = k II 2 1 i x *-= j } £ II II v? = E, n(l) u —1 = 2 > r=0 !{ x u =j) GO - y if S T* r=0 '"'(l) n, rp(l) k j (p, r d) ^ T m where we used the equality: P k t1 1] > t; (r+1) = Pi- X # Tj (1) Pi h 11 > ij (i) (2.52) (2.53) Suppose j =f k; for j = k we have = 1. The equality in (2.53) follows from the strong Markov property as follows. For r = 0 the equality is clear. For r ^ 1 we have ^k = Pt = E k rpO) > rj-i(r+l) k ^ j T, (1) ^ T- r+1) , T, (1) ^ T\ r) + 1 T W + T^ O 7? M ^ T W + rj 1} O 7? ( r ), T fc (1) ^ T W + 1 3 3 T, (1) od (r) ^T| 1} od (r) It (r) rpiX) \ rp{r) M ’ 1 k ^ 1 j + 1 (strong Markov property) Ez, P x .(O Tl 1] > T\ l) , Tl l) > Tj r) + 1 48 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains = E k p, r fc (1) ^ rj 1} , r fc (1) ^ rj r) +1 r fc (1) > if } +1 (induction with respect to r) = IP* if } ^ T d) r™ J, T m (2.54) SinceI T d) ^ T m > 0 it follows by (2.52) that v k > 0. By the same equality and using the fact that P,- T W > T (i) < 1 we see v k < oo. Next we prove the equality: v k = 2 ieS v iPv • Therefore we write 1) 1 h 00 II * 3 II to. ii M ?r X n = j , T™ > n n—1 n—1 = £ P, [X n = j, X n _! = *, T fc (1) ^ n zeS' n=l 00 = X! 2 [p fc [x n = j T n _x], X n _! = i, T fc (1) > ieS n—1 (Markov property) n ieS n=1 £ 2 E t Px„_, [X! = j] , A„_, = i, 2* n !>[*. = j]£P, zeg n=0 2 Pi [A', -j]E X n = i, T, 1 ' - 1 > n ieS 2 Pi [*1 =j]E k ieS Z 1 {*»=<> n=0 1 k Z 1 {*»=<> n= 1 In the last equality of (2.55) we used the equality r (!)_ i k ni(l) 1 k Pfc Z 1 i X n=i} II Z 1 l^n=d 1 3 II o 1_ 1 3 II 1_ (2.55) which is evident for i 4= k and both are equal to 1 for i = k. As a consequence from (2.55) we see that v k = X \ieS v iPij • D 2.15. Corollary. Let the row vector v k := (v k : j e S') be as in equality (2.51) of Theorem 2.If. Then v k is minimal invariant measure in the sense that if 49 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains x = (xj : j e S) is another invariant measure such that #*. = 1. Then xj ^ v*, j e S. x, Proof. We write: = Pkj T X s p s j seS , s=\=k = Pfc = Pfc > P fc Xi = j, r* 11 ;s 1 A'i - j T'" » 1 A'i - j A" # 1 + 2 2 3'S2Ps2SiPsij s±eS, s\^k S2^S + ^ ^S2^525lPsij SlG*S, Si =|=/c Sie5, Sl=|=/c S2^S, S2^k + P fc [x 2 = j, T fc (1) > 2 ] + • • • + P/c X n = j, T, (1) > n (2.56) Upon letting n tend to oo in (2.56) we see that Xj ^ v k . This proves Corollary 2.15. □ qaiteye Challenge the way we run EXPERIENCE THE POWER OF FULL ENGAGEMENT... RUN FASTER. — p RUN LONGER.. RUN EASIER... > 50 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.16. Theorem. Let X be a irreducible Markov chain with transition matrix P = (Pij)(i,j)eSxS ■ The following assertions hold: (a) If any state is non-null recurrent, then all states are. (b) The chain is non-null recurrent if and only if there exists a stationary distribution it or invariant measure. If this is the case, then 1 E k 1 - 1 1—1 and v k = ^ (see (2.51)). (2.57) As a consequence of (2.57) stationary distributions are unique. (b). Proof, (a) The proof of assertion will follow from the proof of assertion (b) Let k be a state which is non-null recurrent. By Theorem 2.12 it follows that the chain is recurrent. By Theorem 2.14 the vector ( v k : j e S) as defined in (2.51) is an invariant vector. Since k is non-null recurrent it follows that 1 0 < Et T (i) < oo, and that the vector — : j e S with 7Tfc = E* T, (i) is a stationary vector. It follows that if the irreducible chain X contains at least one non-null recurrent state, then there exists a stationary distribution. Next suppose that there exists a stationary distribution n := (tt :] : j e S). Then 7Tfe = j e s ^fPj'k f° r n e N. Since the chain is irreducible, and the vector is a probability vector, at least one 7 Tj 0 =)= 0. By irreducibility there exists n e N such that p^l 4= 0, and hence tt/, : A 0, k e S. Consider for any given k e S the 7T ■ \ Then asy. = 1 and by Corollary 2.15 x vector x = ( — : j e S \Kk j e S. It follows that 3 ^ Vj for all E* T;‘ l - 2 >A jeS y jeS nk 1 ^k (2.58) Therefore k is non-null recurrent for all k e S. It follows that if there exists one non-null recurrent vector k s S, then all states in S are non-null recurrent. Altogether this proves assertion (a), and also a large part of (b). From (2.58) the first equality in (2.57) follows. Finally we will show the second equality in (2.57). The vector x — v fc is invariant and, by Corollary 2.15 Hence we obtain, for all positive integers n, 7Ti k V j > °- 0 = 1 ieS k (n) ^ Plk > ffl Kk (■ n ) Pjk- (2.59) 51 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains In (2.59) we choose n in such a way that > 0. By irredncibility this is possible. It follows that (see (2.51)) T7 1_ r T a) i 1 k II II >< _1 o* II i_ II -1 II * _1 7To 7T k i—i Ej 1-1 i—i Vr (2.60) The second equality in (2.57) is the same as (2.60). This concludes the proof of Theorem 2.16. □ Let k be a non-null recurrent state, and suppose that the state is j intercom¬ municates with k. Then both states are non-null or positive recurrent. Next we define the renewal process N k {n ), n e N, as follows: N k {n ) = max jr : < n j . (2-61) Notice the inequalities: rp(Nk{n)) <- ^ ^ rp{Nk{n+\)) and hence T^ n> = n if rn = N k (n). We are interested in the following type of limits: H 71 H 71 lim - V p$ = lim - V P [X t = j \ X 0 = k] n —>oo Ti j J ra —m L 1 J 1 U lim E n —>oo L {Xi=j} | ^-0 Xn = k e=i Put m = N k (n). Then = n, and consequently we see 1 n iVl n h Nk(n ) 1 n m T 771 ( u ) 2 E i U-lg =T (u~l ) + 1 Notice that for j = k we have ^ rn - k n(u) U ~1 + 1 (2.62) (2.63) and consequently, 1 ^ . N k (n) n n t= 1 n Hence, we observe that (see Theorem 2.6) 1 n lim - V 1 71—>00 Ti =r\ = lim N k (n) {X e =k} e=i n—>co n E h 11 1 x„ = k —, Pfc-almost surely. k'k (2.64) 52 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains We also see that N, in) _ 1 T m ,(u) n _ f v i _ Nkij}) \ y v i n Zj n m 2-1 2-1 e=i n m {Xe=j}- (2.65) u-i^ =T ^-i) + i From the strong law of large numbers we get: p( u ) -y Ui k lim — V J] 1{ x t =j} = E oo 771 ^ ^ 1 ; U-l i=T (u 1) + 1 T-i(l) 2 I = k lt=l = 1L- ( 2 . 66 ) From (2.63), (2.64), and (2.65) we obtain: 1 E lim Nj(n) I = j\ Nk(n) 1 ^ ^ ^ n—XX) n («) 1 n lim - V l {x ,=i} n^oo Tl £=1 T m 1 k = lim >oo n m {x e =j} ~ u ~l e=T (u-l) + 1 E h 11 Uo - * (2.67) The equality in (2.67) together with Theorem 2.9 shows the following theorem. 2.17. Theorem. Let the sequence of stopping times (rff^ be defined as in (2.35), and let v( be defined as in (2.66). Suppose that the states j and k intercommunicate and that one of them is non-null recurrent, then the other is also non-null recurrent. Moreover, y n i „,k lim - V 1 {Xe =j} n —>oo n £=1 n E [*f X 0 = J E I x 0 = k and (2.68) -iii -lit lim - Vpg = lim - VP [X e =j \X 0 = k] 71—» 00 Tl j n .—>CD T) L 1 J £= 1 oo n £= 1 E t\ 1) \Xo = j i] e[ t' 1 ' \X 0 = k (2.69) Hence, with i t,- = lim — V* pf], U, = E Tj 1 -* I X 0 = ? , and v% as in equality J n. — >CT) Tl 4—L J J I J ( 1 ) n—>cc n t=\ / > 1 v i (2.66) we have Ttj = — = —. hj l J k 1.1.1. Random walks. In this example the state space is Z, and the process X n , n e N, has a transition probability matrix with the following entries: p,-ij = q, Pi+i,i = P, 0 < p = 1 — q < 1, and p h , = 0, j # i + 1. Such a random walk can be realized by putting X n = V" =f) Sk, where So is the initial state (which may be random), the variables Sk, k e N, k 5= 1, are P-independent of each other and are also P-independent of Sq. Moreover, each variable Sk, k ^ 1, is a Bernoulli variable taking the value +1 with probability p and the value — 1 with probability q. This Markov chain is irreducible: every state communicates 53 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains with every other one. The set of states is closed. The corresponding infinite transition matrix looks as follows: /, q 0 p 0 0 q 0 p 0 0 q 0 • 7 (2 n) Poo = The state 0 has period two Pqq^ = 0, and check transiency (or recurrence) we need to calculate 00 QC / 2 >£ n) = 2 ( 71 = 0 71 = 0 k 2 n n p n q n . In order to ^2 n p n q n . (2.70) By Stirling’s formula we have n\ ~ \Thvnn n e n , which means that n\ lim —j =— n—>G0 v27rnn n e _ = 1 . This e-book is made with SetaPDF QO SETASIGN PDF components for PHP developers www.setasign.com 54 Download free eBooks at bookboon.com Click on the ad to read more Advanced stochastic processes: Part I Renewal theory and Markov chains Since /2 n\ (2 n)\ \f\n to (2 n) 2n e~ 2n 4 n \n J (n!) 2 2nn 2n+1 e~ 2n the sum in (2.70) is finite if and only if the sum (2.71) f ( 4 pgf i \/nn n —1 v (2.72) If p = 1 - ? / then 4pg < 1, and hence the sum in (2.72) is finite, and so the unrestricted asymmetric random walk in Z is transient. However, if p = q = |, then 4 pq = 1 and the sum in (2.72) diverges, and so the symmetric unrestricted random walk in Z is recurrent. One may also do similar calculations for symmetric random walks in Z 2 , Z 3 , and Z d , d ^ 4. It turns out that in Z 2 the 2n-th symmetric transition probability satisfies (for n —► go) — —, 7T7T and hence the sum Xin=o dor”' 1 = oo. It follows that the symmetric random walk in Z 2 is recurrent. The corresponding return probability p () 2 ( "' for the symmetric random walk in Z 3 possesses the following asymptotic behavior: (2 n) 1 / 3 \ 3/2 1 P °° ~2{n) ^ Hence the sum XmLo^oo^ < 00 , and so the state 0 is transient. The 2n-th return probabilities of the symmetric random walk in Z d satisfies (2 n) Cd Poo n ^ G0 > for some constant c a and hence the sum X^o-Poo^ < 00 in dimensions d ^ 3. So in dimensions d ^ 3 the symmetric random walk is transient, and in the dimension d = 1,2, the symmetric random walk is recurrent. We come back to the one-dimensional situation, and we reconsider the return times to a state k e Z: = inf {n 5= 1 : X n = k}. Notice that Sk, k e N, the step sizes which are +1. Also observe that Xj,. = Xj =Q Sj . We consider the are moment generating function Gj : k(s) Ej n(l) S k , 0 ^ s < 1. Observe that on the event jp 1 ' 1 = ooj the quantity s T k ’ has to be interpreted as 0. In addition, we have Fj hi) 7^(1) ^ 7^(1) 1 k ^ 1 k-1 that Ty = T ( jp l + T^ L> o $ (i) , Pj-almost surely, k > j, k, j e Z. Then by the hi) h-1 < °° for k > j, k, j e Z. Then it follows L k — 1 strong Markov property we get Gj,k{s) — E ? p(l) s k = E,- T k-i+ T k 1)o °ui) - 1 . AT < oo = E.- n(l) r(l) od ( 1 ) S^u) . hk < <» 55 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains (Markov property) Ej — Ej n(l) s fc -!Ex ( 1 ) T (l) S k , h-’i < CO rp(l) s T *-' E fc _! S k = E,- rri(l) S k ~l rpiX) S k From (2.73) we see by induction with respect to k that fc-i Gj,k( s ) - k[Gt/+i( s ) - G' 0 ,i(s) fc j . e=j (2.73) (2.74) In the final step of (2.74) we used the fact that the Prdistribution of is the same as the Po-distribution of T \ 1J . This follows from the fact that the variables S m , are independent identically (Bernoulli) distributed random variables. Notice that = 1 + Tp o 0 1 , P 0 -almost surely. Here Tp = inf {n ^ 0 : X n = 1}. Again we use the Markov property to obtain: Gu,i(s) — = sE 0 = sE 0 = sE 0 do) (i) .sT[ Eo Exi Ei = En X* I S, = sEn E rc(O) p(0) , x x = 1] , V = i + sEr + sEn r, [ S T ' ,W ]] [Ea, [4” r , V (notice that T 1 lu; = 0 Pi-almost surely, and P_i-almost surely) = sp + s(?E_i a 1 ! (i) = sp + sqG-i : i(s) = sp + sqG 0t2 (s) = sp + sqG 0t i(s) 2 . (2.75) In the final step of (2.75) we employed (2.74) with j = — 1 and k = 1. From (2.75) we infer 1 — (1 — 4 pqs 2 ) 1 ^ Go As) = 2 qs By a similar token (i.e. by interchanging p and q) we also get 1 — (1 — 4 pqs 2 ) 1 ^ 2 Gi,o(s) — 2 ps (2.76) (2.77) Next we rewrite Go,o( s ) : fjo,o(s) = Eq r rA) s 1q En sEn En s T o 0) ^ I 3l SEn E 'Xi n(l) (on the event {X\ = +1} the equality = Tq 1 ^ holds Pxi-almost surely) = sE 0 ® X1 CC , Xi = 1 + sEq ® X1 rp(A) s 1q 1-1 T—1 1 II 1—1 56 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains = sE 0 [e, [s r o (1) ] , X 1 * l] + sE 0 [e_! [ 5 t o (1) ] , X 1 = -l] = spEi } j + sgE_i } j ((space) translation invariance) = spEi [s T o + sqE 0 [s T l = spG 1>0 (s) + sqG 0>1 (s) (employ the equalities in (2.76) and (2.77)) 1 — (1 — Apqs 2 )^ 2 1 — (1 — A'pqs 2 ) 1 ' 2 = sp 2 7s + sq 2^8 = 1 — (l — 4 pqs 2 y^ 2 . Then we infer P 0 |r 0 (1) < col = lim G 00 (s) = 1 - (1 - 4pg) 1/2 L J stl,s<l = 1 - |1 - 2p\ = 1 - \q-p\ • As a consequence we see that the non-symmetric random walk, i.e. the one with q ¥= p, is transient, and that the symmetric random walk (i.e. p = q = |) is recurrent. However, since E 0 lim. G'q o (s) lim sn,*<l (1 - 5 2 ^ 3/2 = 00 , it follows that the symmetric random walk is not positive recurrent. www.sylvama.com We do not reinvent the wheel we reinvent light. Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges. An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day. OSRAM SYLVAN!A Light is OSRAM 57 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains In the following lemma we prove some of the relevant equalities concerning stopping times and one-dimensional random walks. 2.18. Lemma. Employing the above notation and hypotheses yields the following equalities: (i) The equality = 1 + T^ o holds F^-almost surely for all states k, j e Z. (ii) Fork > j, k, j e Z the equality T^ = T^+T^ o$ ( i) holds Fj-almost k — 1 surely. PROOF. First let us prove assertion (i). Let j and k be states in Z. Then Pfc-almost surely we have 1 + Tj 0 ' 1 o tb = 1 + inf {n ^ 0 : X n od\ = j } = inf {n + 1 : n > 0, X n+1 = j} = inf {n > lX n = j} = TO. (2.78) The equality in (2.78) shows assertion (i). Next we prove the somewhat more difficult equality in (ii). As remarked above we have P,- tA) T 11 1 k ^ ± k —1 dO = p, h-’, < °° Indeed, in order to visit the state k > j the process X n , starting to visit the state A: — 1, and hence P, dO -k -1 di) P,- dd < ® rom j has . Without loss of generality we may and shall assume that in the following arguments we consider the process X n on the event jx^ < ooj. (Otherwise we would automatically have T X l + T^ we write: di) , =Ti 1} . Next on the event {t^ < co} + t£ ] o tf^p = T fc ( i\ + inf jn ^ 1 : X n o # T (p = A:} = inf + n : n> l, X n+T £\ = k } = inf | n > T£\ : X % = k} = t: (i) Assertion (ii) follows from (2.79). Altogether this proves Lemma 2.18. (2.79) □ Perhaps it is useful to prove in an explicit manner that the one-dimensional random walk is a Markov chain. This is the content of the next lemma. 2.19. Lemma. Let {A n : neN,n>l} be independent identically distributed random variables taking their values in Z. Put X 0 = So be the initial value of the process X n , neN, where X n = 2m=o $ m ■ Xwi E, [/ (X 0 , X 1 ,...,X n )] = E[f (X 0 + j, X, d j .X„ f j )] (2.80) for all bounded functions f : Z n+1 —» M, j e Z, neN. Then the equality in (2.80) expresses the fact that the process |(12, S, P j) jeZ , (X n , neN), (d n , n e N), Z j 58 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains is a space homogeneous process, with the property that the distribution of the process {X n +\ — X n ) neN d° es n °t depend on the initial value j. It is also a time-homogeneous Markov chain. PROOF. The first equality say that the finite dimensional distributions of the process (X n ) ngN are homogeneous in space. If f Oo, ■Fl? • • • i *Fn) Q (^1 *Fo i %2 ^1? • * • i ^ J n —l) j then from (2.80) we see that E j [f(X 0 ,X 1 ,...,X n )]=E[f(X 0 + j,X 1 +j,...,X n + j)] = E[p (X 1 -X 0 ,X 2 -X 1 ,...,X n -X n _ 1 )] = Ej [g (. X ! -X 0 ,X 2 -X 1 ,...,X n - X n _i)]. (2.81) The equality in (2.81) shows that the distribution of the process (X n+ \ — X n ) ne pj does not depend on the initial value j. Next we prove the Markov property. Let / be any bounded function on Z. To this end we consider: Ej [f (X n+1 ) | S„] = E [f (X n+1 + j ) | S„] = E [/ (X n+ i — X n + X n + j) | S n ] (the variable X n+ i — X n and the cr-field Sn are IP-independent) = u,~E[u'~f (X n+1 (u') - X n (a/) + X n (cn) + j)] (the P-distribution of X n+ i (u>')—X n (oj r ) neither depends on n nor on the initial value Xq (a/)) = u~E[u/~f(X 1 (u/) - X 0 (u') + X n (u) + j )] (choose X 0 (a;') = j) = u~E[oj’~f(X 1 (X)-j + X n (oj) + j)] = uj~E[u'~f(X 1 (X)+X n (u J ))] = W « E x „ ( „, [V « / (V (V))] = E Xn If (V)] . (2.82) The equality in (2.82) proves Lemma 2.19. □ 2.20. Remark. It would have been sufficient to take / of the form / = 1&, k 6 Z. 2.21. Remark. Lemma 2.19 proves more than just the Markov property for a random walk. It only uses the fact that the increments S n are identically P- distributed and independent, and that the process (AT n ) neN possesses the same Pj-distribution as the P-distribution of the process (X n + j) n6N , j e Z. The proof only simplifies a little bit if one uses the random walk properties in an explicit manner. 59 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 1.1.2. Some remarks. From a historic point of view the references [6, 47, 151] are quite relevant. The references [10, 43] are relatively speaking good accessible. The reference [153] gives a detailed treatment of martingale theory. Citations, like [54, 140, 142, 145] establish a precise relationship between Feller operators, Markov processes, and solutions to the martingale problem. The references [49, 50] establish a relationship between hedging strategies (in mathematical finance) and (backward) stochastic differential equations. Deloitte Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2. Some additional comments on Markov processes In this section we will discuss some topics related to Markov chains and Markov processes. We consider a quadruple m p), (X n , n e N), (# n , n e N), (S, S)}. (2.83) In (2.83) the triple (Q, £F, P) stands for a probability space, fl is called the “sample space”, £F is a a -field on fl. and P is a probability measure on T. The cr-field T is called the cr-field of events. The symbol (S, S) stands for the state space of our process (X n : n e N). In the present situation the state space S is discrete and countable with the discrete cr-field S. Let (Q, T, P) be a probability space and let Xj : fl —> S, j e N = {0,1, 2,...}, be state variables. It is assumed that the cr-field T is generated by the state variables Xj , j e N. Let d k : Tl —*• Tl, k e N, be time shift operators, which are also called time translation operators: Xj of} k = X ]+k , j , k e N. For a bounded ex (Xj : j e M)-measurable stochastic variable F : —*■ M and x e S we write E x [F] = E [F | X 0 = x] E [F, X 0 = a:] P [X 0 = x] We also write Tx - y “ P [x 0 ’=l] X] = Pl[A '‘ = y\= v X = y\ x ° = x \- Here x has the interpretation of state at time 0, and y is the state at time 1. Let Sn, n e be the internal memory up to the moment n. Hence Sn = o (Xj : 0 < j n). 2.22. Theorem. Suppose that (X n :neN) is a stochastic process with values in a discrete countable state space S with the discrete a-field S. The state variables X n , n e N, are defined on a probability space (f2, T, P). Write, as above, T XyV = P [X 1 = y \ X 0 = x\, x, y e S. Then the following assertions are equivalent: (1) For all finite sequences of states (sq, , s n+ i) in S the following iden¬ tity holds: P (X n+ i = s n+ 1 | X 0 = s 0 , •. ■ , X n = s n ) = T SntSn+1 - (2.84) (2) For all bounded functions f : S —> E and for all times n e N the following equality holds: E [/ (X n+ i) | Sn] = [/ (V)] P -almost surely, (2.85) (3) For all bounded functions / 0 ,.... f k op S and for all times neN the following equality holds P -almost surely: E [/o(X n )/ 1 (X re+1 )... f k (X n+k ) | Sn] = [/o(X 0 )/ 1 (X 1 )... f k (X k )] ; (2.86) (4) For all bounded measurable functions F : (Q, T) —>■ M (stochastic vari¬ ables) and for all n e N the following identity holds: E[Fod n \S n ]=E Xn [F] P -almost surely ; (2.87) 61 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains (5) For all bounded functions f : S —► E and for all (Sn) n6 N -stopping times t : 12 — > [0, co] the following equality holds: E [/(X r+ 1 ) | S T ] = Ex t [/(Xi)] F-almost surely on the event {r < go} ; ( 2 . 88 ) (6) For all bounded measurable functions F : (Cl, £F) —> E (random vari¬ ables) and for all stopping times r the following identity holds: E [F o | S r ] = E Xt [F] P -almost surely on the event {r < oo} . (2.89) Before we prove Theorem 2.22 we make some remarks and give some explana¬ tion. 2.23. Remark. Let Cl = 5 ,H . equipped with the product cr-field, and let X :] : Cl —*• S be defined by Xj(cu) = ojj where u = (coo, ..., Wj ,...) belongs to Cl. If dk : Cl —*• Cl is defined by dk (coo,..., Uj ,...) = (u>k, ■ ■ ■, ojj+k, ...), then it follows that Xj o dk = Xj + k . 2.24. Remark. Instead of one probability space (12,T, P) we often consider a family of probability spaces (0. T, P x ) x6iS .. The probabilities P x , x e S, are determined by E, [F] = E [F | X„ = x] = E p^° = ° j" 1 , (2-90) Here F : Cl —*• E is T-lBp-nieasurable, and hence by definition it is a random or stochastic variable. Since P x [A] = E x [lu], dej, and P x [12] = E x [1] = 1, the measure P x is a probability measure on ‘J. 2.25. Remark. Let F : Cl —> E be a bounded stochastic variable. The variable E\- n [F] is a stochastic variable which is measurable with respect to the cr-field a (X n ), i.e. the cr-field generated by X n . In fact we have E XnH [F] = E [F | X 0 = X n (co)] = = E [a/ i—> F (a/) x l{x 0 E[F, X 0 = X n (uj)\ P [X 0 = X n (uj)\ =X n (uj)} (<F)] • (2.91) If we fix u e 12, then in (2.91) everything is determined, and there should be no ambiguity any more. 2.26. Remark. Fix neN. The cr-field S n is generated by events of the form { (Xq, X\ , . . . , -X n ) (s0; Si j • • • ) S n )} . Here (so> si, • • •, s n ) varies over S n+ 1 . It follows that 9n = cr{{(X 0 ,X 1 ,...,X n ) = (s 0 ,si,...,a;„)} : (s 0 , s u ..., s n ) e = {{(X 0 ,X 1 ,...,X n )eR}: = cr(X 0 ,X i,...,X n ). (2.92) The cr-field in (2.92) is the smallest cr-field rendering all state variables Xj, 0 ^ j < n, measurable. It is noticed that S n c T. 62 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.27. Remark. Next we discuss conditional expectations. Again let A : 12 —> M. be a bounded stochastic variable. If we write Z = E [F | S„], then we mean the following: (1) The stochastic variable Z is Sn-®R- m easurable. This a qualitative as¬ pect of the notion of conditional expectation. (2) The stochastic variable Z possesses the property that E [F, A] = E [Z. A] for all events A e £j n . This is the quantitative aspect of the notion of conditional expectation. Notice that the property in (2) is equivalent to the following one: the stochastic variable Z satisfies the equality E [F | A] = E \Z | A] for all events A e S„. 2.28. Remark. Let S be a sub-cr-field of T. The mapping F •—» E [F | S] is an orthogonal projection from L 2 (Q, T, P) onto L 2 (Q, 3, P). Let F e L 2 (fl, T, P), and put Z = E [F | 3], In fact we have to verify the following conditions: (1) Z e L 2 (f2, S, P); (2) If G e L 2 (fl, g,P), then the following inequality is satisfied: E[|F-F| 2 ] < E [|F — G\ 2 ] . This claim is left as an exercise for the reader. For more details on conditional expectations see Section 1 in Chapter 1. 2.29. Remark. Next we will give an explicit formula for the conditional ex¬ pectation in the setting of a enumerable discrete state space S. Let Sn = a (Xq,X±, ... ,X n ) where Xj : Q —> S, 0 ^ j < n, are state variables with a discrete countable state space S. In addition, let F : fl —> M be a bounded stochastic variable. Then we have E [F | Sn] = ^ E [F | X 0 = i 0 ,... ,X n = i n ] l{x 0 =i 0 } • • • 1 {x n =i n }- (2.93) Writing the conditional expectation in (2.93) in an explicit manner as a function of uj yields E [F | Sj ( W ) = ^ E [F | X 0 = i 0 ,..., X n = i n ] l{x 0 =j 0 }(w) • • • l{x n =i n }(u). (2.94) From (2.93) and also (2.94) it is clear that the conditional expectation E [F | gj is S n -measurable. 2.30. Remark. Put S = Soo = cr (A”o,..., X n ,...) = a (X) where X : ^ S N is the variable defined by X (cu) = (Xo(cij),..., X n (u >),...), u> e fl. Then S = Soo (2.95) = {{AeF}: Fis measurable with respect to the product a-field on ,S N } . 2.31. Remark. Let r : fl —> N u {cxd} be a random variable. This random variable is called a stopping time relative the filtration (S )t ) rt6F [, or, more briefly, 63 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains r is called a (S n ) ne N _s ^°PPi n S time, provided that for every k e N an event of the form {r < k) is Sat measurable. The latter property is equivalent to the following one. For every k e N the event {r = k } is Sat measurable. Note that {t = k} = {t ^ k) \ {t < k — 1}, k e N, k ^ 1 , and {r < k) = u j =0 {t = j). From these equalities it follows that r is a (S n ) ner! -stopping time if and only if for every k e N the event {r = k} is Sat measurable. 2.32. Remark. Let B be a subset of S. Important examples of stopping times are t r = inf {k ^ 0 : e B } on u* =0 { X ^ e B } and oo elsewhere; t r = inf [k ^ 1 : Ah, e B } on u* =1 { X e B } and oo elsewhere. (2.96) Similarly we also write = inf {k ^ s : X}. e B} on the event u* =s {X}. e B\, and Tg = oo elsewhere. The time t r is called the hrst income time, and is called the hrst hitting time, or the hrst income time after 0. We also notice that = min {A; ^ 1 : I^eB} on u“ =1 { X\ e Bj and = oo on r\f =1 {Xg e S\B}. In addition: 1 + o •d 1 = (} l R . 2.33. Remark. Again let r : fl —> N u {oo} be a (Sn) n6 Fr s t°PPi n g time. The cr-held St containing the information from the past of r is dehned by St = {A e T : A n {t ^ k} e Sfc} = a(X JAT : je N) (2.97) where X jAT (uj) = X jATiuj) (uj), u e Q. SIMPLY CLEVER SKODA We will turn your CV into an opportunity of a lifetime Do you like cars? Would you like to be a part of a successful brand? We will appreciate and reward both your enthusiasm and talent. Send us your CV. You will be surprised where it can take you. 64 Send us your CV on www.employerforlife.com Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.34. Remark. Let F be a stochastic variable. What is meant by F o $ k and F o d T on the event {r < go}? Here k e N, and r is a (S n ) n6N -stopping time. For F = n; =0 fj (^j) we write: n n f ° = n u (A) ° a= n h (*>+*). 3=0 3=0 and on the event {r < go} F ° «r = n Si (A) ° «T = n fi ( X > + • j =o j =o 2.35. Proposition. Let (ril 1 ) (2.98) be a sequence of square matrices with (x,y)eSxS positive entries, possibly with infinite countably many entries (when S is count¬ able, not finite). Put n+1 7£„ -T‘„ = 7$, T”„- 2 | | / .... s n+i = V- (2.99) si,...,s n e5 i=l The equalities in (2.99) are to be considered as matrix multiplications. Fix 1 ^ n\ < ■ ■ ■ < nk ^ n, and let the measure space (S k , (x)^ =1 S, l\li,...,n k ,n+i,y)> (x,y)eSxS,neN be determined by the equalities: r . n n fj ( S i) d 9°nZ,n k ,n+l, Sn+1 (« 1 , ■ • • , «fc) 7=1 k+1 - S nbhWLfe), /,EP(S,S), (• si,...,sp)eS k j=l ( 2 . 100 ) where (so>Sn+i) = (x,y) e S x S, no = 0, and nk+i = n + 1. Then /or ever?/ 1 ^ Jo ^ R, and fj e L 00 (R, £), 1 < / ^ n, this family satisfies the following equality: r> n fi fj ( S i) ^R.°,n,n+l, Sn+1 (si, • • • , S n ) (2-101) J * n i=l P n = I ]^I fj ( S j) dpfj n+1 (si, • • • , Sjo-l, s jo+l) • • • > S n) T's ; ,- 0 _ 1 ,s jo+1 ' j=l,jVjo where Tf y = 2 zeS^^Tz^y for all (x,y) e S x S (matrix multiplication). Let 1 < ni < • • • < n k < n, and put Tnl..,n k ,n +1 (#0 X R) = 1 B „ (^hn’L.^n+l,,/ (#) , B 0 E§, Be ® k §. yeS ( 2 . 102 ) Suppose that the matrices ( Tx() , n 6 N ; are stochastic. Then the (x,y)eS x S' measures in (2.102) do not depend on n+1. Moreover, the following assertions are equivalent: 65 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains (a) The family of measure spaces {(S fc+1 ,® fc+1 S,/z°’ 1 a ;... >nfc>n+1 ) : 1 n x < ••• < n k < n, n e N} (2.103) is a consistent family of probability measure spaces. (b) The family of measure spaces defined in (2.100) is consistent. (c) For every neff and (x,y) e S x S the equality T x )jJ = Tf y holds. Suppose that the family in (2.103) is a consistent family of probability spaces. Then the corresponding process {(tl,?,F x ) xeS ,(X n : neN), (d n , n e N), (S, S)} . (2.104) is a Markov chain if and only if for (x,y) e S x S and n, me N the following matrix multiplication equality holds: This means that rji(n-\-m) \ 1 rp(ri) m(m) \ 1 rpn rjim 1 x,y / j 1 x,z 1 z,y / j 1 x^z 1 z,y zeS zeS (2.105) P’s [X 0 e B 0 ,..., X n e B n \ = 1 -b 0 {x) iTf* ^ n+l (Bi x • • • x B n ) , for Bj e §, 0 < j < n, and that the family in (2.104) possesses the Markov property if and only (2.105) holds. In addition, we have T x n y = P x [. X n = y], x, y e S; i.e. the quantities T x n y represent the n time step transition probabilities from the state x to the state y. 2.36. Theorem. Let the notation be as in Theorem 2.22. The following asser¬ tions are equivalent: (1) For every s e S, for every bounded function f : S —> R, and for all n 6 N the following equality holds F s -almost surely: E. [/ (X.+i) | Sn] - E x „ [/ (X,)]. (2.106) (2) For every bounded function f : S —* M, and for all n 6 N the following equality holds P -almost surely: E [/ (X n+1 ) | Sn] = Ex, [/ (X x )]. (2.107) 2.37. Remark. From the proof it follows that in Theorem 2.36 we may replace the stochastic variable / (Xi) by any bounded stochastic variable Y : O —*• R. At the same / (X n+ i) = / (Xi) o D n has to be replaced by Y o d n . 2.38. Remark. Theorem 2.36 together with Remark 2.37 shows that through¬ out in Theorem 2.22 we may replace the probability P with P s for any s e S. Consequently, we could have defined a time-homogeneous Markov chain as a quadruple m P s ) seS , (X„ : n 6 N), (0 fc , k e N), (S, S)} satisfying the equivalent conditions in Theorem 2.22 with P 5J for all s e E, instead of P. 66 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains Proof of Theorem 2.36. (1) => (2) Let s e S, f : S — M a bounded function, and neN. From (2.106) we infer E s [f (X n+1 ) ,A]=E a [E Xn [f (X 1 )], A] for all A e S n . (2.108) From (2.108) we infer E [f (X n+l ),A,X 0 = s\ = E [E Xn [/ (X ra+ i)], A, X 0 = s] for all A e $ n , and seS. (2.109) By summing over s e S in (2.109) we obtain E[f(X n+1 ),A] = E[E Xn [f(X n+1 )],A] for all ^4 e Sn- (2.110) From (2.110) the equality in (2.107) easily follows. (2) => (1) Let / : S —> K. and neNbe such that (2.107) holds. Then, since all events of the form {X 0 = s}, s e S, belong to Sn, (2.107) implies that (2.109) holds for / and hence by dividing by P [X 0 = s], for s e S, we obtain (2.108). Hence (2.106) follows. All this completes the proof of Theorem 2.36. □ MAERSK I joined MITAS because I wanted real responsibility The Graduate Programme for Engineers and Geoscientists www.discovermitas.com Real work International opportunities Three work placements a I was a construction supervisor in the North Sea advising and helping foremen solve problems Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains The following theorem is similar to the formulation of Theorem 2.36 but now with stopping times and having remark 2.37 taken into account: 2.39. Theorem. Let the notation be as in Theorem 2.22, and letr:Cl^> [0, go] be a (5n) ne ^- s topping time. The following assertions are equivalent: (1) For every s e S, and for every bounded stochastic variable Y : Cl —> K. the following equality holds P s -almost surely on the event {t < oo}; E a [Yot? r |g r ]=E* T [y]. (2.iii) (2) For every bounded stochastic variable Y : Cl —> R. the following equality holds P -almost surely on the event {r < go}; E[hotf r | S T ]=E Xr [Y]. (2.112) 2.40. Remark. Let r : Cl —» Nu{oo} be a (S n ) neN -stopping time, and let A e S r . Let m e N u {go}. Put r m = mla\A + rl a- Then r m is a (S n ) neN -stopping time. If P[A] < 1 and m = go, then r m = oo on the event Ct\A which is non-negligible. 2.41. Definition. Let Cl be a set and let S be a collection of subsets of Cl. Then S is called a Dynkin system, if it has the following properties: (a) Cl e S; (b) if A and B belong to S and if A => B, then A\B belongs to S; (c) if (A n : n e N) is an increasing sequence of elements of S, then the union Un=i A n belongs to S. In the literature Dynkin systems are also called A-systems: see e.g. [3]. A 7r-system is a collection of subsets which is closed under finite intersections. A Dynkin system which is also a 7r-system is a a-field. The following result on Dynkin systems, known as the 7T-A theorem, gives a stronger result. 2.42. Theorem. Let M be a collection of subsets of Cl, which is stable under finite intersections, so that M is a ir-system on Cl. The Dynkin system generated by M coincides with the a-field generated by M. PROOF. Let T) (M) be the smallest Dynkin-system containing M, i.e. D (M) is the Dynkin-system generated by M. For all A e T) (M), we define: T{A) := {B e V (M) : AnBeV (M)}. then we have (1) if A belongs to M, M c r(A), (2) for all A e M, T(A) is a Dynkin system on Cl. (3) if A belongs to M, then T> (M) c= T(A), (4) if B belongs to CD (M), then OVf c T(B), (5) for all B e D (M) the inclusion, D (OVf) c T(B) holds. It follows that CD (M) is also a 7 r-system. It is esay to see that a Dynkin system which is at the same time a 7r-system is in fact a a-field (or cr-algebra). This completes the proof of Theorem 2.42. □ 68 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.43. Theorem. Let Cl be a set and let M be a collection of subsets of it, which is stable (or closed) under finite intersections. Let IK be a vector space of real valued functions on Cl satisfying: (i) The constant function 1 belongs to K and 1 a belongs to K for all A e M; (ii) if (f n : n e N) is an increasing sequence of non-negative functions in K such that f = sup neN f n is finite (bounded), then f belongs to K. Then K contains all real valued functions (bounded) functions on that are ct(M) measurable. Proof. Put T) = [A ^ Cl ; 1^ e K}. Then by (i) Cl belongs to T) and T) ^ M. If A and B are in D and if B ^ A, then B\A belongs to D. If ( A n :neN) is an increasing sequence in T>, then 1 u a„ = sup n 1 a„ belongs to T) by (ii). Hence T) is a Dynkin system, that contains M. Since M is closed under finite intersection, it follows by Theorem 2.42 that T> 3 <r(M). If / ^ 0 is S n2 n l{f^j 2 -ny Since the n subsets {/ ^ j2~ n }, j, n e N, belong to cr(M), we see that / is a member of K. Here we employed the fact that cr(M) ^ D. If / is a (M)-measurable, then we write / as a difference of two non-negative a (M)-measurable functions. □ The previous theorems (Theorem 2.42 and Theorem 2.43) are used in the fol¬ lowing form. Let Q be a set and let (Si, §i) ieI be a family of measurable spaces, indexed by an arbitrary set /. For each i e I, let M t denote a collection of subsets of Si, closed under finite intersection, which generates the a- field S,;, and let f) : A S, be a map from Q to ,5). In this context the following two propositions follow. 2.44. Proposition. Let M be the collection of all sets of the form (~] ieJ ff 1 (Ai), Ai e M i, i e J, J <= I, J finite. Then M is a collection of subsets of Ll which is stable under finite intersection and <r(M) = a (fi : i e I). 2.45. Proposition. Let K be a vector space of real-valued functions on Cl such that: (i) the constant function 1 belongs to K; (ii) if ( h n : n e N) is an increasing sequence of non-negative functions in K such that h = sup n h n is finite (bounded), then h belongs to K; (iii) K contains all products of the form Idiej U; ° /i, J — I, J finite, and Ai e Mi, i e J. Under these assumptions K contains all real-valued functions (bounded) func¬ tions in a(fi : i e /). The theorems 2.42 and 2.43, and the propositions 2.44 and 2.45 are called the monotone class theorem. 69 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains In the propositions 2.44 and 2.45 we may take S{ = S', M* the collection of finite subsets of Si, and /* = X i: i e I = N. 3. More on Brownian motion Further on in this book on stochastic processes we will discuss Brownian motion in more detail. In fact we will consider Brownian motion as a Gaussian process, as a Markov process, and as a martingale (which includes a discussion on Ito calculus). In addition Brownian motion can be viewed as weak limit of a scaled symmetric random walk. For this result we need a Functional Central Limit Theorem (FCLT) which is a generalization of the classical central limit theorem. 2.46. Theorem (Multivariate Classical Central Limit Theorem). Let (Q,T, P) be a probability space, and let {Z n : neN} be a sequence of ¥-independent and ¥-identically distributed random variables with values in W 1 in L 1 (fl, T, P; M d ). Let n = E[Zi], and let D be the dispersion matrix of Z\ (i.e. the variance- covariance of the random vector Z\). Then there exists a centered Gaussian (or multivariate normal) random vector X with dispersion matrix D such that the sequence Z\ + • • • + Z n — Ufa converges weakly (or in distribution) to a centered random vector X with dis¬ persion matrix D as n —> go. The latter means that lim E [/ ( Z n )] = E [/ (Z)] n —>00 for all bounded continuous functions f : —» M. Notice that by a non-trivial density argument we only need to prove the equality for all functions / of the form f(x) = e ~ l ^ x ^, x e £ e M d . Next let us give a (formal) definition of Brownian motion. 2.47. Definition. A one-dimensional Brownian motion with drift /i and diffu¬ sion coefficient a 2 is a stochastic process {X(t) : t ^ 0} with continuous sam¬ ple paths having independent Gaussian increments with mean and variance of an increment X(t + s) — X(t) given by sp = E [X(t + s) — X(f)] and sa 2 ■ [(A(t + s) — A(t)) 2 ], s, t ^ 0. If X 0 = x , then this Brownian is said to start at x. A Brownian motion with drift /.i = 0, and a 2 = 1 is called a standard Brownian motion. One of the problems is whether or not such a process exists. One way of resolv¬ ing this problem is to put the Functional Central Limit Theorem at work. Let us prepare for this approach. Let {Zj : j e N} be a sequence of centered inde¬ pendent identically distributed real valued random variables in L 1 (Q. T, P) with variance a 2 = E [Z 2 ] . For example these variables could be Bernoulli variables taking the values +a and —a with the same probability Put So = Z 0 = 0, 70 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains S n = Zi + «• • , n e N, n ^ 1. Define for each scale parameter n ^ 1 the stochastic process X^ n \t) by c. . vL nt J 7 X^\t) = ^ = ^ 0 . (2.113) V n V n Here [nt\ is the integer part of nt, i.e. the largest integer k for which k < nt < k + 1. The it is relatively easy to see that E [A (n) (f)] = 0, and Var (A (n) (f)) = E [X (n) (f) 2 ] = to 2 . (2.114) Then the classical CLT (Central Limit Theorem) implies that there exists a process { X(t ) : t ^ 0} with the property that for every rn e Id, for every choice (ti ,.... t m ) of rn positive real numbers, and every bounded continuous function / : R m —> C the following limit equality holds: lim E [/ (X™(t))] = E [/ (X(t))] . (2.115) The equality in (2.115) says the finite-dimensional distributions of the sequence of processes \X (n) (t) : t ^ 0} n( _ N converges weakly to the finite-dimensional distributions of the process {X(t) : t ^ 0}. This limit should then be one¬ dimensional Brownian motion with drift zero and variance o 2 . A posteriori we know that Brownian motion should be P-almost surely continuous. How¬ ever the processes [X^ n >(t) : t ^ 0} neN have jumps. It would be nice if we were able to replace these processes which have jumps by processes without jumps. Therefore we employ linear interpolation. This can be done as follows. We introduce the following interpolating sequence of continuous processes: X {n \t) = ^ + (nt - [nt\) Z[n *i +1 , t > 0. (2.116) Jn Jn T ll rrui mm+1 Let m and n be positive integers. Then on the half open interval —,- |_ n n S ~ the variable X n (t ) is constant in time t at level —^, while X n (t) changes linearly from S n at time t n TM , $712 + 1 — to n Jn S n + J m +1 n at time t m + 1 n n jx( n )(t) : t ^ 0 (2.117) } 72EN It can be proved that the sequence of stochastic processes converges weakly to Brownian motion with drift /i and variance <r 2 . This is the contents of the following FCLT (Functional Central Limit Theorem). The following result also goes under the name “Donsker’s invariance principle”: see, e.g., [15] or [42], 2.48. Theorem (Functional Central Limit Theorem). Let^X^ n \t) : te [0,T]j, ne N, and {X(t) : te [0,T]} be stochastic processes possessing sample paths which are P -almost surely continuous with the property that the finite-dimen¬ sional distributions of the sequence \X^(t) : t e [0,T] 1 converge weakly to l J 12EN 71 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains those of {X(t) : te [0,T]}. Then the sequence \x^ n \t) : te [0,Til con- v. J neN verges weakly to {X(t) : t e [0, T]} if and only if for every e > 0 the following equality holds: lim sup P 'T 0 neN sup X {n \s) -X {n \s) > £ 0^s,£^T, \s—t\^S = 0 . (2.118) This result is based on Prohorov’s tightness theorem and the Arzela-Ascoli characterization of compact subsets of C'[(l. T], 2.49. Theorem (Prohorov theorem). Let (P n : neN) be a sequence of proba¬ bility measures on a separable complete metrizable topological space S with Borel a-field S. Then the following assertions are equivalent: (i) For every £ > 0 there exists a compact subset K £ of S such that P n [K e ] ^ 1 — e for all neN. (ii) Every subsequence of (P n : n e N) has a subsequence which converges weakly to a probability measure on (S, S). A sequence (P n ) n satisfying (i) (or (ii)) in Theorem 2.49 is called a Prohorov set. Theorem 2.48 can be proved by applying Theorem 2.49 with P n equal to the P-distribution of the process | XW(t) 0 < t < T }■ Because achieving your dreams is your greatest challenge. IE Business School’s Master in Management taught in English, Spanish or bilingually, trains young high performance professionals at the beginning of their career through an innovative and stimulating program that will help them reach their full potential. Choose your area of specialization. Customize your master through the different options offered. Global Immersion Weeks in locations such as London, Silicon Valley or Shanghai. Because you change , we change with you . www.ie.edu/master-management mim.admissions@ie.edu f # In YnTube ii Master in Management • 72 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.50. Theorem (Arzela-Ascoli). Endow C'[0, T] with the topology of uniform convergence. A subset A of C[ 0 , T] has compact closure if and only if it has the following properties: (i) sup |w( 0 )| < go; ujeA (ii) The subset A is equi- continuous in the sense that lim sup sup |cn(s) — w(t)| = 0 . NO OsSs,tCT, |s-t|«:<5 o>eA From (i) and (ii) it follows that sup sup |w(s)| < go, and hence A is uniformly ujeA se[0,T] bounded. The result which is relevant here reads as follows. It is the same as Theorem T.8.4 in Bhattacharaya and Waymire [15]. 2.51. Theorem. Let ( P n ) n be a sequence of probability measures on (7[0,T]. Then ( P n ) n is tight if and only if the following two conditions hold. (i) For each p > 0 there is a number B such that P n [u e C[0,T] : |cn(0)| > B] < p, n = 1 , 2 , ... (ii) For each z > 0, p > 0, there is a 0 < S < 1 such that Pn ueC[0,T] : sup \u(s) — u(t)\ ^ z 0^s,£^T, \s— 1\^5 n = l, 2 , PROOF. If the sequence (P n ) n is tight, then given p > 0 there is a compact subset K of C ([ 0 ,T]) such that P n (K ) > 1 — p for all n. By the Arzela-Ascoli theorem (Theorem 2.50), if B > sup w6X |co(0)|, then P n [a; 6 C[0,T] : | W (0)| > B] < P n [K c ] < 1 - (1 -p)=p. Also given z > 0 select 5 > 0 such that sup sup |cn(s) — cj(f)| < £• Then ujeK \s— 1\^5 w e C[0, T] : sup \u(s) — u(t)\ ^ s 0 \s—t\^S < P n [.A c ] < p for all n ^ 1. The converse goes as follows. Given p > 0, first select B using (i) such that P n \uj e C ([0,T]) : |co(0)| < B] ^ 1 — ^p, for n 5 = 1 . Select <5 r > 0 using (ii) such that Pr uj e C ([0, T]) : sup 0^s,£^T, \s—t\^S IjO(s) — U;(t)| < 5= 1 — 2 (r+ 1 i ?7 for n 5= 1 . Now take K to be the uniform closure of OO f f~l < uj e C ([0, T]) : |cn(0)| < B, sup r= l I 0^s,t^T,\s—t\^S u(s)-u(t )I < A . Then P n (K) > 1 — 77 for n ^ 1 , and K is compact by the Arzela-Ascoli theorem. This completes the proof Theorem 2.51. □ 73 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains For convenience of the reader we formulate some limit theorems which are rel¬ evant in the main text of this book. The formulations are taken from Stirzaker [ 126 ]. For proofs the reader is also referred to Stirzaker. For convenience we also insert proofs which are based on Birkhoff’s ergodic theorem. Define Sn = s: ~o Xk, where the variables \Xk} keN a L 1 (Q, IF, P) are independent and identically distributed (i.i.d.). Then we have the following three classic results. 2.52. Theorem (Central limit theorem, standard version). If E [X^] = y and 0 < var (X^ = a 2 < co, then lim P 71—>00 S n nfi (ncr 2 ) 1 / 2 < x = <F(a;), where 'F(x) is the standard normal distribution, i.e. <h(x) = \Z2tt J_ r e"^ 2 dx. PROOF. Let / : M —» C be a bounded (7 2 -function with a bounded second derivative. Then by Taylor’s formula (or by integration by parts) we have f(y) = /(0) + yf'(0) + iy 2 /"(0) + J (1 -s)y 2 {f'(sy) - f"(0)} ds. (2.119) Put Y n k = - ;4 ‘ ■ Inserting y = Y n k into (2.119) yields ’ (ryn / (UU = /(o) + Y n , k f( 0) + ir„y"( 0) + £ (1 - s) Yl k {/" (sY„, k ) - /"( 0)( ds. ( 2 . 120 ) Then we take expectations in (2.120) to obtain: [/ (Y„, t )] = m + ^/"(0) + £ (1 - s) E [y„ 2 it {/" (sY„, k ) - /"(«)}] ds. ( 2 . 121 ) E Put M = n f(l - s)E K, (1 - Jo L )] ds, t and choose f(y) = e~ lty . Observe that, uniformly in t on compact subsets of M, lim n ^oo£ n (t) = 0. Then, since the variables Y U: k, 1 < k < n, are i.i.d., from (2.121) we get E \e ~ itY ».*1 = 1 - — + t ^l . From (2.122) we infer ( 2 . 122 ) E = Efc=i Li. k (2.123) Let Y : Q —*• M be a standard normally distributed random variable. From the properties of the sequence [s n (t)} n and (2.123) we see that, for every 0 < R < co, lim sup IE n ^°° \t\*ZR ( e ■*!£= l Y n, k - e -i* Y } 74 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains = lim sup \ ;e 1- J w HO 7 03 n ^°° \t\^R 1 = lim sup \ n ^°° \t\^R 1 Je g—it 2fc = l ^n,k n 0 . (2.124) The conclusion in Theorem 2.52 then follows from (2.124) together with Levy’s continuity theorem: see Theorem 5.42, and Theorem 5.43 assertions (9) and ( 10 ), □ 2.53. Theorem (Weak law of large numbers). If E [W] = /a < co, then for all £ > 0 , lim P 71 —► 00 Sn -h n > s 0 . For a proof of the following theorem see (the proof of) Theorem 5.60. It is proved as a consequence of the (pointwise) ergodic theorem of Birkhoff: see Theorems 5.59 and 5.66, and Corollary 5.67. "I studied English for 16 years but... ...I finally learned to speak it in just n six lessons Jane, Chinese architect ENGLISH OUT THERE Click to hear me talking before and after my unique course download Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.54. Theorem (Strong law of large numbers). The equality g lim — = /i, holds F-almost surely (2.125) n—>co n for some finite constant //,, if and only i/E[|Ah|] < co, and then fi = E[Ah]. Moreover, the limit in (2.125) also exists in L 1 -sense. We will show that Theorem 2.53 is a consequence of Theorem 2.54. Proof of Theorem 2.53. Let £ > 0 be arbitrary, and let {X k } k and /i be as in Theorem 2.53. Then P S n n/i n n (2.126) By the Id-version of Theorem 2.54 it follows that the right-had side of (2.126) converges to 0. This shows that Theorem 2.53 is a consequence of Theorem 2.54. □ The central limit theorem is the principal reason for the appearance of the nor¬ mal (or “bell-shaped”) distribution in so many statistical and scientific contexts. The first version of this theorem was proved by Abraham de Moivre before 1733. The laws of large numbers supply a solid foundation for our faith in the useful¬ ness and good behavior of averages. In particular, as we have remarked above, they support one of our most appealing interpretations of probability as long¬ term relative frequency. The first version of the weak law was proved by James Bernoulli around 1700; and the first form of the strong law by Emile Borel in 1909. We include proofs of these results in the form as stated. As noted above a proof of Theorem 2.54 will be based on Birkhoff’s ergodic theorem. 2.55. Remark. The following papers and books give information about the central limit theorem in the context of Stein’s method which stems from Stein [124]: see Barbour and Hall [9], Barbour and Chen [8], Chen, Goldstein and Shao [31], Nourdin and Peccati [101], Berckmoes et al [13]. This is a very inter¬ esting and elegant method to prove convergence and give estimates for partial sums of so-called standard triangular arrays (STA). It yields sharp estimates: see the forthcoming paper [14], 4. Gaussian vectors. The following theorem gives a definition of a Gaussian (or a multivariate nor¬ mally distributed) vector purely in terms of its characteristic function (Fourier transform of its distribution. 2.56. Theorem. Let (f2,3 r , P) be a probability space, and let X = (Ah, ..., X n ) be an E n -valued Gaussian vector in the sense that there exists a vector p := (/ii,..., n n ) e E n and a symmetric square matrix a := (c r yfc)” fc=1 such that the characteristic function of the vector X is given by E [e“* <€ ’ x) ] = for all £ = (&,..., f n ) e R n . (2.127) 76 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains Then for every 1 ^ j ^ n the variable Xj belongs to L 2 (fl, IT, P), /j 7 = E [Xj\, and (2.128) <Tj,k = cov (Xj, X k ) = E [(Xj - E [X,]) (X k - E [. X k ])]. Proof. Put Y = X — /i, and fix e > 0. Then the equality in (2.127) is equivalent to E [e = e 2 £i£k a j,k f or a n £ = (£ 1? ... 5 £ n ) (2.129) From Cauchy’s theorem and the equality e-^in 2 = 2lT£) n Jr o-dvT) P -h'"' 2 ( \Z2ne ) From (2.129) and (2.130) we infer: E e 2 s |??l dr/ I r® _i ^j-oo 6 2 srl 2?? dr/ = 1 we obtain (V^) e 2,7,1 dr}. (2.130) ~ p -i{£.,Y) p -\e\Y\ 2 ~ - 1 f E ~ -i<£+V^T>~ (V 2 ^) n Jr- O e 2 1^1 dr} (employ (2.129) with £ + yejy instead of f ) (V^Tl) JM n (2.131) Next we take 1 ^ l\, (-2 ^ n, and we differentiate the right-hand side and left-hand side of (2.131) with respect to and the result with respect ly, • In addition we write a negative sign in front of this. Then we obtain: E e-^Yt^e-tW 1 f e-2 J]R n (V2n y a £iA + a hA ~2'm,k=i(ij+Y^Vj)(A+\^VkYj,k p -2\v\ X fe + V^Vj) 0=1 a jA + a h,j dg. (2.132) Inserting f = 0 into (2.132) yields: E FqF, 2 e-^l y l 2 Jr (V2ny a h/2 + a hA o-h £ Tv,k=i r ljrik‘Tj,k p -h\ 1 l \ 2 — £ 0=1 a jA + a h,j dr}. (2.133) First assume that t\ = £2 = C Then the left-hand side of (2.133) increases to E [Y 2 ], and the right-hand side increases to _ n e — 2 1 7 ?! dr/ oyr = cqy if £ (V27t) jRn decreases to zero. Consequently, Y^ e L 2 (Q, T, P) and E [Y) 2 ] = cqy, 1 < £ ^ n. 77 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains It follows that Yt belongs to L 1 (12, T. P), and that we also have that Yf L Y( 2 e L 1 (12, T. P). By applying the same procedure as above we also obtain that E [Y h Y h ] = ^ + 2 ^ (2.134) In (2.134) we employed the symmetry of the matrix ( 04 ^)™ ^ v Again we fix 1 < I < n, and we differentiate the equality in (2.131) with respect to & to obtain iE ' e -i(ix) Yie -W? 7 J-xn f dr] e~^l k =^ + Y~evj) + e ~h\ (v 2n) jRn 2 fe + V^Vj) 3 = 1 &j,e + G i. (2.135) In (2.135) we set £ = 0, and we let e J, 0 to obtain E \Y(\ = 0, and hence Xt e L 1 (Q, ff, P) and E [A/] = ]_i(. This completes the proof of Theorem 2.56. □ 5. Radon-Nikodym Theorem We begin by formulating a convenient version of Radon-Nikodym’s theorem. For a proof the reader is referred to Bauer [10] or Stroock [130]. 2.57. Theorem (Radon-Nikodym theorem). Let (O, T, /x) be a a-finite measure space, and let u be a finite measure on T. Suppose that v is absolute continuous relative to //,. i.e. ji(B) = 0 implies u(B) = 0. Then there exists a function f e L 1 (12, T. p) such that v(B) = / dp for all B e T. In particular the function f is T-measurable. The following corollary follows from Theorem 2.57 by taking T = “B, p the measure P confined to 23, and v(B) = E [X, B\, B e 23. 2.58. Corollary. Let (12, A, P) be a probability space, and let 23 be a subfield (i.e. a sub-a-field) of A. Let X be a stochastic variable in L 1 (12, A, P). Then there exists a 23 -measurable variable Z on 12 with the following properties: (1) (qualitative property) the variable Z is 23 -measurable; (2) (quantitative property) for every Be® the equality E \Z, B] = E [X, B] holds. The variable Z is called the conditional expectation of X, and is denoted by Z = E [A J 23]. The existence is guaranteed by the Radon-Nikodym theorem. 6. Some martingales Let FI be a locally compact Hausdorff space which is second countable, and let {(12, T, F x ) , (A (t),t > 0), (d t , t > 0), (E, £)} (2.136) 78 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains be a time-homogeneous strong Markov process with right-continuous paths, which also have left limits in the state space E on their life time. Put S(t)f(x ) = E x [/ (X(£))], / e Cq(E), and assume that S(t)f e Cq(E) whenever f e Cq(E). Here a real or complex valued function / belongs to Cq{E) provided that it is continuous and that for every £ > 0 the subset {x <e E : \f(x)\ > e} is compact in E. Let the operator L be the generator of this process. This means that its domain D(L ) consists of those functions / e Cq{E ) for which the limit S(t)f -f Lf = lim t|0 t exists in G'o( E ) equipped with the supremum norm, i.e. Il/H^ = sup xs£; \ f{x)\, f 6 Cq(E). The Markov property of the process in (2.136) together with the right continuity of paths implies that the family {S(t) : t ^ 0 } is a Feller, or, more properly, a Feller-Dynkin semigroup. ( 1 ) The semigroup property can be expressed as follows: S {ti + t 2 ) = S (ti) S (t 2 ), ti, t 2 ^ 0, 5(0) = /. (2) Moreover, the right-continuity of paths implies lim S(t)f(x) - BmE, [/ (V(i))] - E* [/ (X (0))] - f(x), f e Co(E). (3) In addition, if 0 < / < 1 , then 0 < S(t)f < 1 . A semigroup with the properties ( 1 ), ( 2 ) and (3) is a called a Feller, or Feller- Dynkin semigroup. In fact, it can be proved that a Feller-Dynkin semigroup {S(t) : t ^ 0} satisfies iim ||5(s)/-5(t)/|| oo = 0 , t> 0, / 6 Cq(E). s —s>0 Let {S(t) : t ^ 0} be a Feller-Dynkin semigroup. Then it can be shown that there a exists a Markov process, as in (2.136) with right-continuous paths such that S(t)f(x) = E* [/(X(£))], / e Cq(E), t ^ 0. For details, see Blumenthal and Getoor [ 20 ]. Similar results are true for states spaces which are Polish; see, e.g., [146]. Let t > M(t), t ^ 0 , be an adapted right-continuous multiplicative process, i.e. M( 0) = 1 and M(s)M(t ) o = M(s + t), s, t ^ 0. Put 5jw(f)/(x) = E x [M(t) f (X(t))], / g Cq(E), t ^ 0 . Assume that the operators Sm( t) leave the space Cq(E) invariant, so that 5 m( f)/ belongs to Cq{E ) whenever / g Cq{E). Then the family {Sm{ t) : t ^ 0} has the semigroup property 5 m(5 + 1) = S M (s)S M (t), s,t> 0 , and lim ao S M (t)f(x) = f(x), t ^ 0 , / g C 0 (E). If, in ad¬ dition, for every / g Cq(E) there exists a S > 0 such that sup 0 ^ t ^ 5 ||5m (^)/|| 00 < co, then ton II S M (t)f ~ f loo = 0, / g C 0 (E). (2.137) Moreover, there a exists a closed densely defined linear operator Lm such that L M f = C 0 (E)- lim - 1 (2.138) for f e D ( Lm ), the domain of Lm • If M(t ) = 1 , then Lm = L. 79 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.59. Proposition. The following processes are ¥ x -martingales: t ~ M(t)f (X(t)) - M(0)f(X(0)) - f M(s)L M f (X(s) ds ), Jo t> o, feD{L M ), (2.139) s - M(s)E x{s) [M(t - s)f (X(t - s))] , O^s^t, feC 0 (E), (2.140) s i-^>-M(s)Ex( s )[M(t — s — u)p(u,X(t — s — u),y)], 0 ^ s ^ t — u. (2.141) In (2.141) It is assumed that there exists a “reference” measure m on the Borel field £ together with an density function p(t,x,y), ( t,x,y ) e (0, go ) x E x E such that E x [f (X(t))] = §p(t,x,y) f(y) dm(y) for all f e Co(E ) and for all x e E and all t > 0. From the semigroup property it follows that p(s + t, x, y) = ,x, z)p(t,z,y)dm(z ) for m-almost all y e E. Assuming that m(0 ) > 0 for all non-empty open subsets of E, and that the function (t, x, y ) > p(t, x, y) is continuous on (0, 00 ) x E x E, it follows that the equality p(s + t, x, y) = $p(s , x, z)p(t, z, y)dm(z) holds for s, t > 0 and for all x, y e E. AACSB ACCREDITED Excellent Economics and Business programmes '\&r university of groningen www.rug.nl/feb/education “The perfect start of a successful, international career.” CLICK HERE to discover why both socially and academically the University of Groningen is one of the best places for a student to be 80 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains The following corollary is the same as Proposition 2.59 with M = 1. 2.60. Corollary. The following processes are P x -martingales: t - f (*(£)) - f(X( 0)) - f Lf (X(s) ds), t> 0, feD(L), (2.142) Jo s - Ej (s) [f (X(t - s))], 0 < s < t, f e C 0 (E), (2.143) s i—> p (£ — s, X(s), y), 0 < s < t. (2.144) Like in Proposition 2.59 in (2.144) it is assumed that there exists a “reference” strictly positive Borel measure m such that for a (unique) continuous density function p(t,x,y) the identity E x [f (X(t))] = §p(t,x,y) f(y)dm(y) holds for all f e Co(E ) and for all x e E and all t > 0. 2.61. Lemma. Let the continuous density be as in Proposition 2.59, and let z e E. Then the following equality holds for all 0 ^ s < t and for all y e E: E z (p{t - s,X(s),y)] = p(t, x, y). (2.145) Proof of Lemma 2.61. Let the notation be as in Lemma 2.61. Then by the identity of Chapman-Kolmogorov we have E ; z [p(t - s,X(s),y)\ = J p(s,z,w)p(t - s,w,y)dm(w) =p(t,x,y). (2.146) The equality in (2.146) is the same as the one in (2.145), which completes the proof of Lemma 2.61. □ Proof of Proposition 2.59. First let / belong to the domain of L M , and let £2 > C ^ 0. Then we have E x \m (t 2 ) f (X (t 2 )) - M(0)/ (X(0)) - f M(s)L M f {X{s)) ds \ T tl Jo - M (P) f [X (t 2 )) + M(0)f (A"(())) + f 1 M(s)L M f (X(s)) ds Jo = E x M (£,) (t 2 - h) f (X (t 2 - h)) - M(0)f (X(0)) - J Q <2 ^ M(s)L m f{X(s)) ds^j od tl | T tl (Markov property) — M (£1) Expp M (£ 2 - ti) f (X (£ 2 - ti)) - M(0)/ (X(0)) \ 2 11 M(s)L M f(X(s)) ds Jo (definition of the operator 5 m(£); put z = X (£ 1 ), and t = t 2 — t±) = M (£,) (5 m (t) f(z) - E z [M(0)/(X(0))] - £ S M (s)L M fiz) ds 81 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains = M (h) (s M 0 1 ) f(z) - E z [M(0)f (X(0))] - j- s S M (s)f(z ) ds^j = M (h) ( S M (t) f(z) - E z [M(0)f (X(0))j - S M (t)f(z) + S M (0)f(z)) = 0. (2.147) The equalities in (2.147) show the equality in (2.139). Let / 6 Cq(E) and t > 0. In order to show that the process 3 - M(s)E x(a) [M(t - s)f (X(t - s))] is a P ;1 ,-inartingalc we proceed as follows: M(s)E x(s) [M(t-s)f(X(t-s))] (Markov property) = M(s)E x [M(t - s) o d s f (X(t - s)) o $ s | 3F S ] = E x [M(s)M(t - s ) o d s f (X(t)) | T s ] = E x [M(t)f (X(t)) | T s ] . (2.148) It is clear that the process in (2.148) is a martingale. This proves that the process in (2.140) is a martingale. A similar argument shows the equality: M(s)Ex(s) [M(t - s - u)p (u, X(t - s - u), y)] = E x [M (t — u)p (u, X(t — u),y) | T s ] , 0 < s < t — u. (2.149) Again it is clear that the process in (2.149) as a function of s is a P^-martingale. Altogether this proves Proposition 2.59. □ Proof of Corollary 2.60. The fact that the processes in (2.142) and (2.143) are P^-martingales is an immediate consequence of (2.139) and (2.140) respectively by inserting M(p) = 1 for all 0 < p < t. If M(p) = 1 for all 0 < p <t, then by Lemma 2.61 we get M (s)Ex( s ) [M(t - s - u)p (u, X(t - s - u ), yj\ = Ex( s ) [p(u,X(t - s - u),y )] =p(t- s,X(s),y) . (2.150) On the other hand by the Markov property we also have: Ex( a ) [P (u,X(t - s - u), y )] = E x [p (u,X(t-u),y) | Tj . (2.151) As a consequence of (2.150) and (2.151) we see that the process in (2.144) is a martingale. This completes the proof of Corollary 2.60. □ 2.62. Remark. In general the process s >—► p (t — s, X(s), y), 0 < s < t, is not a closed martingale. In many concrete examples we have lim p(t — s,X(s),y) = s]t,s<t 0, P x -almost surely, on the one hand, and E x [p(t — s,X(s),r/)] = p(t,x,y ) > 0 on the other. For an example of this situation take d-dimensional Brow¬ nian motion. By Scheffe’s theorem it follows that the P ; ,-martiiigale s i—> p{t — s,X(s),y), 0 < s < t, can not be a closed martingale. If it were, then there would exist an T r mcasurable variable F(t) = lim s -| ~ :S<t p (t — s, X(s),y) with the property that pit — s,X(s),y) = E x \_F{t) | T s ]. Since F(t) = 0, P x - almost surely, this is a contradiction. 82 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains In the following corollary we consider a special multiplicative process: M(s) = 1{t>s}> where T is a terminal stopping time, i.e. T = s + T o d s , P^-almost surely, on the event {T > s} for all s > 0 and for all x e E. 2.63. Corollary. The following processes are P x -martingales: t - 1 {T>t} f (X(t)) - l {T>o} f(X(0)) - f AT Luf (X(s) ds ), Jo t> o, feD(L M ), (2.152) s ^ 1{t> s }E X ( s ) [f (X(t - s)), T > t - s], 0 < s < t, f e C 0 (E), (2.153) s ^ 1 {T>S} (p(t - s,X(s),y) - E x(s) [p(t-s- T,X(T),y ), T < t - s]) , 0 ^ s < t. (2.154) In (2.141) it is assumed that there exists a “reference” measure m on the Borel field £ together with an density function p{t , x, y), (t , x, y) e (0, go) x E x E such that Ea, [/ (X(f))] = J p (t, x, y) f(y) dm(y) for all f e C${E) and for all x e E and all t > 0. American online LIGS University is currently enrolling in the Interactive Online BBA, MBA, MSc, DBA and PhD programs: ► enroll by September 30th, 2014 and ► save up to 16% on the tuition! ► pay in 10 installments/2 years ► Interactive Online education ► visit www.ligsuniversity.com to find out more! Note: LIGS University is not accredited by nationally recognized accrediting agency by the US Secretary of Education. More info here. Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains It is noticed that the definition of L M f(x ) is only defined pointwise, and that for certain points x e E the limit L M f(x).- lim- 1 - does not even exist. A good example is obtained by taking for T the exit time from an open subset U: T = Tu = inf {s > 0 : X(s) e E\U}. If the ip \t < t\ lim —-—--= 0 for all x e U, then Lm/(x) = Lf(x) for x e U. Proof of Corollary 2.63. It is only (2.154) which needs some explana¬ tion; the others are direct consequences of Proposition 2.59. To this end we fix 0 < u <t. Then by (2.141) the process: s h-> l {r>s} Ex( s ) \p(u,X(t - s - u),y ), T > t - s - u] is a martingale on the closed interval [0 ,t — u \. Next we rewrite Ex-(a) [p (u, X(t - s - u),y), T > t - s - u\ = E X (s) [p(u,X(t - s- u),y )] -E x(s) \p{u,X{t - s - u),y ), T < t - s - u] (2.155) (the process p >—> p(t — s — p,X(p),y) is P z -martingale with z = X(s); put u = t — s in the first term, and u = t — s — T in the second term of the right-hand side of (2.155)) = E X ( S ) [p(t - s,X(0),y)] - Ex(,) [p(t- s- T,X(T),y ), T ^ t - s - u]. (2.156) By letting u { 0 in (2.156) and using (2.141) of Proposition 2.59 we obtain that the process in (2.154) is a P ; ,.-martingale. This completes the proof of Corollary 2.63. □ Next let {(a T, P x ), > 0 ), (0 t ,t > 0 ), (R d , S Rd )} be the Markov process of Brownian motion. Another application of martingale theory is the following example. Let U be an open subset of W ! with smooth enough boundary dU {C l will do), and let / : dU M be a bounded continuous function on the boundary dU of U. Let a : U M be a continuous function such that u(x) = f(x) for x e dU and such that A u{x) = 0 for x e U. Let tu be the first exit time from U: Tu = inf |.s > 0 : B(s) e Then u(x) = E x [f (B (■ t v )) : tjj < go] + lim E* [a (B (£)) : t v = oo]. (2.157) 00 Notice that the first expression in (2.157) makes sense, because it can be proved that Brownian motion is P x -almost surely continuous for x e W l . The proof uses the following facts: stopped martingales are again martingales, and the processes t - / (B(t)) - f (5(0)) - \ J A/ (5(s)) ds, f e C b (R d ), A/ e C b (M d ) , (2.158) 84 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains are martingales. The fact that a process of the form (2.158) is a martingale follows from (2.142) in Corollary 2.60. It can also be proved using the equality j t Pd (t, X , y) = ^\p d (t, X , y) (2.159) where Pd ( t,x,y ) = 1 (2tt t)° I x-y \ 2 -e 2t A proof of (2.158) runs as follows. Pick t 2 > h > 0 en a function / e C& (R d ), such that A/ also belongs to C'i, (M d ). Then we have: E, / (B (t 2 )) - f (B (0)) - i J ’ A/ (B(s)) ds | 3-, - / (B (ti)) + / (B (0)) + 1A/ (B(s)) * (/ (B (*2 - ti)) - / (B (0)) - i “ A/ (B(s)) = E, (Markov property of Brownian motion) ds oA % E Bit i) / (B ih ~ ti)) - / (B (0)) - i P “ A/ (B(s)) ds (put z = B (t\), and t = f 2 — ti) = E z [/ (5(0)] - E z [/ (5(0))] - 1 £ E z [A/ (5(s))] ds = E z [/ (5(0)] - E z [/ (5(0))] -If f p d (s, z, y) A/ (y) dy ds Z Jo J]R d = E 2 [/ (B(t))] - Ej [/ (B(0))] - lim iff p d (s, z, if) A/ (») dy £ 1° 2 J £ J R d (integration by parts) = E, [/ (B(i))] - Ej [/ (B( 0 ))] - lim f f i A, Pi (s, z, y) / (y) dy is £ 1° Je Jr<* 2 (use the equality in (2.159)) = E z [/ (5(t))] - E z [/ (5(0))] - lim f f ^ Pd (s, z, y) f (y) dy ds £ 1° Je jRd dS (interchange integration and differentiation) = E 2 [/ (5(0)] - E z [/ (5(0))] - lim f f p d (s, 2 , y) f (y) dy ds •" J £ ds J Rd (fundamental rule of calculus) = E s [/(B(t))]-EJ/(B(0))] 85 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains -lip ( Pd (t, z, y) f (y) dy- p d (e, z, y) f (y) dy = E z [f (B(t))] - E z [f (5(0))] - E, [f (B(t))] + limE, [f (5(e))] e|0 -limE 2 [/(B( £ ))]-E,[/(B(0))]=0. (2.160) From Doob’s optional sampling theorem it follows that processes of the form t~f(B(TuAt))-f(B(0))-^ T J At Af(B(s))ds, feC b (R d ), (2.161) / e C b (M d ), A/ 6 C b (M d ), are P x -martingales for x e U. We can apply this property to our harmonic function u. It follows that the process t h-> u(B (ju a t)) — u (5(0)) — A u (5(s)) ds = u(B (tjj a t)) — u (5(0)) 2 J 0 (2.162) is a martingale. Consequently, from (2.162) we get u(x) = u (5(0)) = E x [u (5 (t v a t))] = E x [u (5 (txj a t )), Tu t] + E x [u (5 (t v a t)), t v > t] = E x [u (5 (tu)) , t v < t\ + E x [u (5(f)), t v > t] (2.163) In (2.163) we let t —*• oo to obtain the equality in (2.157). A cate-Lucent www.alcatel-lucent.com/careers What if you could build your future and create the future? One generation’s transformation is the next’s status quo. In the near future, people may soon think it’s strange that devices ever had to be “plugged in.” To obtain that status, there needs to be “The Shift". Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains 2.64. Proposition. Let t > Mi(t ) and t > M 2 (t) be two continuous martin¬ gales in L 2 (12, T, P) with covariation process t > (Mi, M 2 ) (f), so that in partic¬ ular the process t •—» Mi(t)M 2 (t) — (Mi, M 2 ) (t) is a martingale in L 1 (fi, T, P). T/iera t/ie process t ~ (Mi(t) - Mi(s)) (M 2 (t) - M 2 (s )) - (Mj, M 2 ) (f) + (Mi, M 2 ) (s), is a martingale. t ^ s, (2.164) In fact by Ito calculus we have the following integration by parts formula: (M 1 (f)-M 1 (s))(M 2 (f)-M 2 (s)) = f (Mi(p) - Mi(s)) dM 2 (p) + f (M 2 (p) - M 2 (s)) dMi(p) J S J S + (M\, M 2 ) (f) — (Mi, M 2 ) (s), t ^ s. (2.165) Proof of Proposition 2.64. Fix t 2 > t x ^ s. Then we calculate: E [(Mj (f 2 ) - Mi (s)) (M 2 (t 2 ) - M 2 (s) | T tl )] - E [(Mi, M 2 ) ( t 2 ) - (Mi, M 2 ) (s) | T tl ] - (Mi (ti) - Mi (s)) (M 2 (ti) - M 2 (s)) + (Mi, M 2 ) (ti) - (Mi, M 2 ) (s) = E [(Mi (t 2 ) - Mi (s)) (M 2 (t 2 ) - M 2 (s)) | T tl ] - E [(M l5 M 2 ) (t 2 ) - (M 1} M 2 ) (ti) | T tl ] - (Mi (h) - Mi (s)) (M 2 (ti) - M 2 (s)) = E [(Mi (t 2 ) - Mi (ti) + Mi (ti) - Mi (s)) (M 2 (t 2 ) - M 2 (ti) + M 2 (h) - M 2 (s)) -(Mi,M 2 ) (t 2 ) + (Mi, M 2 ) (ti) |T tl ] - (Mi (ti) - Mi (s)) (M 2 (ti) - M 2 (s)) = E [(Mi (t 2 ) - Mi (ti)) (M 2 (t 2 ) - M 2 (h)) | T tl ] - E [(Mi, M 2 ) (t 2 ) + (Mi, M 2 ) (ti) | J h ] + E [(Mi (h) - Mi (s)) (M 2 (t 2 ) - M 2 (h)) | T tl ] + E [(Mi (t 2 ) - Mi (ti)) (M 2 (ti) - M 2 (s)) j J tL ] + E [(Mi (h) - Mi (s)) (M 2 (ti) - M 2 (s)) | T tl ] - (Mi (ti) - Mi (s)) (M 2 (ti) - M 2 (s)) = E [Mi (t 2 ) M 2 (t 2 ) - (Mi, M 2 ) (t 2 ) + (Mi, M 2 ) (ti) - Mi (ti) M 2 (ti) \ T tl ] -E [Mi(s) (M 2 (t 2 )-M 2 (ti)) \? tl ] -E[(Mi(t 2 )-Mi(ti))M 2 (s)\% 1 ] + E [(Mi (h) - Mi (s)) (M 2 (ti) - M 2 (s)) | T tl ] - (Mi (h) - Mi (s)) (M 2 (h) - M 2 (s)) = 0. (2.166) 87 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Renewal theory and Markov chains In the final step of (2.166) we employed the martingale property of the following processes: t i— ► Mi(t)M 2 (t) — (Mi, M 2 ) (i), 1 1— ► and 1 1—> M 2 (t). This completes the proof of Proposition 2.64. □ In the past four years we have drilled * 81,000 km A That's more than twice around the world. Whn am wp? fHSHHHH P We are the world's leading oilfield services company. Working 1 globally—often in remote and challenging locations—we invent, design, engineer, manufacture, apply, and maintain technology to help customers find and produce oil and gas safely. Who are we looking for? We offer countless opportunities in the following domains: ■ Engineering, Research, and Operations ^ ■ Geoscience and Petrotechnical ■ Commercial and Business A ^ If you are a self-motivated graduate looking for a dynamic career, apply to join our team. What will you be? careers.slb.com Schlumberger 88 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales CHAPTER 3 An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In this chapter of the book we will study several aspects of Brownian motion: Brownian motion as a Gaussian process, Brownian motion as a Markov process, Brownian motion as a martingale. It also includes a discussion on stochastic integrals and Ito’s formula. 1. Gaussian processes We begin with an important extension theorem of Kolmogorov, which enables us to construct stochastic processes like Gaussian processes, Levy processes, Poisson processes and others. It is also useful for the construction of Markov processes. In Theorem 3.1 the symbol Qj, J Q I, stands for the product space ^J = YljeJ Qj endowed with the product a- field ‘Jj. By saying that the system {(Clj, Tj, Pj) : J c /, J finite} is a projective system (or a consistent system, or a cylindrical measure) we mean that [p& A] = Pj, (di) W) where A e T/ 2 , and where ^ J\ ^ /, -h finite. The mapping p J fi , J 2 Q Ji, is defined by p J fi 2 (cOj) jeJi = ( UJ j)j e j 2 • practice this means that in order to prove that the system {(Gj, Tj,Pj) : J <= /, J finite} is a projective system indeed, we have to show an equality of the form (j 0 ^ J): [B x %„] = Pj [5], B e Jj. The following proposition says that under certain conditions a cylindrical mea¬ sure in fact is a genuine measure. 3.1. Theorem (Extension theorem of Kolmogorov). Let {(nj,Tj,Pj) : Jc/, J finite} be a projective system of probability spaces (or distributions). Suppose that each 0, is a metrizable and a-compact Hausdorff space endowed with its Borel field Ai. Then there exists a unique probability measure Pj on (f h,Ai), such that P/ \Pj e A] = P 7 (pj\A)) = Pj(A) (3.1) for every J c= / ; J finite, and for every A e Aj. For an extensive discussion on Kolmogorov’s extension theorem see, e.g., the Probability Theory lecture notes of B. Driver [40]. These lecture notes include 89 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales a discussion on standard Borel spaces and on Polish spaces. The Kolmogorov’s extension theorem is also valid if the spaces 0* are Polish spaces, or Souslin spaces which are continuous images of Polish spaces. For more details see Ap¬ pendix 17.6 in [40]. The reader may also consult [21] or [137]. In Theorem 7.4.3 of [21] the author shows that finite positive measures on Souslin spaces are regular and concentrated on a-compact subsets. Bogachev’s book contains lots of information on Souslin spaces. In fact much material which is presented in this book, can also be found in the lecture notes by Bruce Driver. A proof of Kolmogorov’s extension theorem is supplied in Section 4 of Chapter 5: see (the proof of) Theorem 5.81. Next we recall Bochner’s theorem. 3 .2. Theorem. (Bochner) Let : M n —> C be a continuous complex function, that is positive definite in the sense that for all r eN r ^ A fc A,<p (£*-£') >0, (3.2) k,£= 1 for all Ai,..., A r e C and for all ,... ,£f e R n . Then there exists a unique non-negative Borel measure ji on R n such that its Fourier transform JexpH «,.))*.(*) is equal to <p(£) for In particular p(M n ) = </?(0). 3.3. Example. Let, for every i e I, Pj, i e I, be a probability measures on Q, ( and define Pj on flj, J <= /, J finite, by Pj(A) = P^ ® • • -®Pj n (A), where A be¬ longs to Aj and where J = (j i,... ,j n ). Then the family {P j : Jcl, J finite} is a consistent system or cylindrical measure. 3.4. EXAMPLE. Let cr:/x/—>-Mbea symmetric (i.e. cr(i,j) = cr(j, i) for all i, j in I) function such that for every finite subset J = (j i,..., j n ) of I the matrix ( a (h J))i jeJ 7S positive-definite in the sense that 0, (3.3) i,jeJ for all €ju ■ ■ ■, f j n e M. In the non-degenerate case we shah assume that the inequality in (3.3) is strict whenever the vector (£ 7l , ■ ■ ■ ,£j n ) is non-zero. De¬ fine the process ( i,uj ) >-> Xfiui) by Xfiui) = u>i, where w e O; = R 7 is given by uj = (. Let p = (/q) e R 7 be a map from / to M. There exists a unique probability measure P on the cr-held on if generated by (X, ) iel with the following property: E ^exp J] tjX^ j = exp j exp ^ <r(b j)&£jj . (3.4) This measure possesses the following additional properties: E (Xj) = Uj, j e I, and cov (A7*, Xf) = a{i, j), i,j e I. (3.5) 90 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Notice that ^ £ u £ v cov (Xj u , Xj v ) ^ 0 whenever £ 1 ,... belong to K. For u,v =1 a proof of this result we shall employ both Bochner’s theorem as well as Kol¬ mogorov’s extension theorem. Therefore let J = (ji ,..., j n ) be a finite subset of /, let Afc, 1 < k < r, be complex numbers and let £ k , 1 < k < r, be vectors in R n = R J . Put A^ = A*, exp (?’ X" = i C/ 7 /,) an d ^ U be an orthogonal matrix with the property that the matrix {UaU~ l {u,v)) v av=l has the diagonal form (s\ 0 ... 0 \ /rf\ . We also write (rj{,... = U (i UaU-\u,v )) 0 4 \0 0 0 4/ VC/ We may and do suppose that the eigenvalues si,..., s m , m ^ n, are non-zero and the others (if any) are 0. Then we get r _ ( \ n Yi A a-A/- exp - j J] a(j u ,j v ) (4 - 4) (4 - 4) M=1 \ lt,P = l X exp (-1 2 (4 - 4) \ u=l = J] A lV ex P (-i J (UaU -1 ) («w) W “ d9 M=1 V = 2 A Cr ex P 2 s « W - d9 2 ) M=1 V U=1 / m 9 2 «« M - 2 Aa 4 exp k/= 1 S AA 1 /c,£=l n=l (V 2 ^) m nr. — 1 1 = 1 5 U J J day...day exp 2 (4 - 4 ) ay j exp ^ 2 (V / 27r)" i nr=i^I'"I < day ... day / m 2 A fc ex p (* 2 ^ k —1 \ it=l exp M v C) V > 0 . (3.6) From Bochner’s Theorem 3.2 it follows that there exists a probability measure Ilj on M J such that, for all (eR", exp i V £ u x u dU V 41 / 91 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales = exp ( ~ l 2 J eX P ( 2 <*(ju,jv)£u& V u= 1 / \ u,v= 1 (3.7) on a, = n jeJ n j by Define the probability measure irj on = [ y. eJ ^ \\ ji > ) jn / / 'J \ j > where B is a Borel subset of R J . The collection (Elj, Aj. Pj) is a projective system, because let J 1 : = {jo} u J be a subset of I, which is of size 1 + size J = 1 + n and let B be a Borel subset of R J . The Fourier transform of the measure B h-> II ji [R x B] is given by the function: , • • •, £j n ) ^ exp -z V £jXj n j, [R x dx] JrJ V jtj J 2 tjXj n r [dy x dx] d<= T ) -f j Jm j Jm exp : ex P ( 2 ) exp ( “h 2 V jeJ’ J V i,jeJ> ■ eX P ( -*2^i ) exp ( ~\ 2 V&M&i \ jsJ J \ i,jeJ / ■ f C '-r./b.O | I heW /t4 ^ / i Maastricht University Join the best at the Maastricht University School of Business and Economics! gjpj* • 33 rd place Financial Times worldwide ranking: MSc International Business • 1 st place: MSc International Business • 1 st place: MSc Financial Economics • 2 nd place: MSc Management of Learning • 2 nd place: MSc Economics • 2 nd place: MSc Econometrics and Operations Research • 2 nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier ‘Beste Studies' ranking 2012; Financial Times Global Masters in Management ranking 2012 Maastricht University is the best specialist university in the Netherlands (Elsevier) Master's Open Day: 22 February 2014 www.mastersopenday.nl j Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In the previous formula we used the equality £j 0 = 0 several times: J' = ,/u {jo}. It follows that Ilj [5] = II ji [E x B]. An application of the extension theorem of Kolmogorov yields the desired result in Example 3.4. Suppose that the matrix (cr(j u , j v ))u V =i non-degenerate (i.e. suppose that its determinant is non-zero) and let (a(u,v))™ v=1 be its inverse. Then P ((X jl ,...,X h )sB) (det a) 1 / 2 (2n) n / 2 exp H dx 1 . . . dx n lB(xi, ..., x n ) 1 n N - 2 a ( u , v ) ( x u - du) 0 X v - dv) u.v—1 (3.8) Equality (3.8) can be proved by showing that the Fourier transforms of both measures in (3.8) coincide. In the following propositions (Propositions 3.5 and 3.6) we mention some elementary facts on Gaussian vectors. Gaussian vectors are multivariate normally distributed random vectors. 3.5. Proposition. Let (12,T, E) be a probability space and let X 1 : 0 >—> E n % i = 1, 2, be random vectors with the property that the random vector X(cu) : = (X 1 (o;), X 2 (tv)) is Gaussian in the sense that (n = n\ + n<i) E A M=i (3.9) where the matrix cr(k,£) ki=1 is positive definite and where (/.q,... ,/r n ) is a vec¬ tor in M n . The vectors X 1 and X 2 are ¥-independent if and only if they are uncorrelated in the sense that E (XlXj) = E (Xl) E (X 2 ) (3.10) for all 1 < i < n\ and for all 1 < j < n 2 . Proof. The necessity is clear. For the sufficiency we proceed as follows. Put {X\X 2 ) = (X u ..., X ni ,X ni+u X ni+n2 ). Since the vectors X 1 and X 2 are uncorrelated (see (3.10)), it follows that n 1 2 a ( k ^)t,k& + a (M)6c6- (3.11) k,£= 1 k,£— 1 k/=n\ + l From (3.9) it follows that E ( exp ( -i J] Z,kX k k=1 n i ni+ri2 = E j exp ( -i J] f k X k ) ) E j exp | —i J] 6 kX k k=n± + l k —1 and hence that the random vectors X 1 and X 2 are independent. (3.12) □ 93 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.6. Proposition. Let (f2,£F, P) be a probability space. (a) Let Q : M n be a linear map. If X : Q —► M m is a Gaussian vector, then so is QX. (b) A random vector X : Ll —*• M n is Gaussian if and only if for every (el n the random variable uj > (£,X(u)) is Gaussian. PROOF, (a) A random vector X is Gaussian if and only if the Fourier trans¬ form of the measure B >—> P (X e B) is of the form f ex P G* (f, h) ( a ^ ■ By a standard result on image measures the Fourier transform of the measure B i—► P ( QX e B ), where X : Q —► W 1 is Gaussian and where B is a Borel subset of K m , is given by £ - E [exp (-* <£, QX))] = E [exp (-* (Q% X))] = exp (-* (Q*£, p)) exp (aQ*£, Q*£)^J ■ (3.13) This proves (a). It also proves that the dispersion matrix of QX is given by QcrQ*. (b) For the necessity we apply (a) with the linear map Qx := (£,x), x e R n , where £ e E n is fixed. For the sufficiency we again fix £ e E". Since Y := (£, X) is a Gaussian variable we have E (exp (-i (f, X))) = E (exp (-W)) = exp (-fE(F)) exp ^-^E (Y - E(F)) 2 = exp (—i (£, /i}) exp (cr£,£)^ , (3.14) where // = E(X) and where a(k,£) = cov(X k ,X e )=E(X k -E(X k ))(X e -E(X e )). This completes the proof of (b). □ 3.7. Theorem. Let a : I x / —*• M be a positive-definite function and let p : I —> M be a map. There exists a probability space (Q, T, P) together with a Gaussian process ( t,u> ) > X t (uf) = X(t,cu), t e I, u e Ll, such that E(X t ) = [i t and such that cov(X s , X t ) = a(s, t ) for all s, t e I. Proof. The proof is essentially given in Example 3.4. □ We conclude this section with the introduction of Brownian motion and Brow¬ nian bridge as Gaussian processes. First we show that the function a : [0, go) x [0, go) -> E, defined by cr(u,v) = min(^, v), u, v e [0, oo), and, for t ^ 0 fixed, the function a t : [0,t] x [0,i] —> M, defined by cr t (u,v) = tmm(u,v) — uv, u , v e [0,i], are positive definite. 94 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.8. Proposition. The functions a(u,v) = mm(u,v), u, v e [0, oo), a t (u,v) = tmin(u, v) — uv, u, v e [0, t\, and ctr(u, v) = - exp (— |u — v\)j u, v e R ; are positive definite. In addition, the function ao(u,v) defined by <Jq(u, v) = - exp(— (u + w)) (exp (2min(w, v)) — 1), u, v 5= 0, is positive definite. > Apply now REDEFINE YOUR FUTURE AXA GLOBAL GRADUATE PROGRAM 2015 redefining /standards £ 95 ^0 Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Proof. Let 0 = s 0 < si < s 2 < s 3 < • • • < s n < t and let Ai,..., X n be complex numbers. The following identities are valid: 2 t^Sj — Sj- 1 ) 2 3 = 1 ^2 Afc(t Sk) k—j n n n (t-SjW-Sj- 1) n n n r = 2 2 2 (* - Sfcj (* - *0 | j=lk\=iko = j ^ j = l Aji=J /c 2 =j n n min(fci,fc 2 ) = 2 2 2 Afc i Afc 2 fci=l ^2 = 1 j —1 n n Sj — i t — Sj t — i 5 o — i t Sj t Sj_ 1 ip &ki ) ip $k2 ) = 2 "7 (fa,fa) fc 1= lfc 2 = l s mm(/ci,/c 2 ) n n n = 2 2 Afei Afc 2 ^min(fci,fc 2 ) (t ^max(fci,fc 2 )) 2 Afei A& 2 , s^ 2 ) (3.15) fcl=lfc 2 = l /ci ,fc 2 = l and hence the function a t is positive definite. Since n n n ^ ^ j AjAfet min(sj, s k ) ^ j AjA^(j^(sj, ^ ] A j,fc=i i,fc=i i=i it follows that the function cr is positive definite as well. In order to prove that the function is positive definite we first notice that the Fourier transform of the function t i—► exp (— |t|) is given by r oo rco e-^e^dt = 2 cos (ft) e“*dt J-oo Jo = 2Re f = 2Re : Jo 1 1-zf 1 + f 2 Hence upon taking the inverse Fourier transform we obtain: (3.16) I e -|*-s| 2 J_ f°° exp (if(t-s)) 2tt J_ m f 2 + 1 (3.17) Let Ai,..., X n be complex numbers and let Si,... ,s n be real numbers. From (3.16) and (3.17) it follows that 2 Wx exp (— | s k - 8e \) k,i =1 i r i 27T J.qo f 2 + 1 n 2 A fc exp (ifs fc ) k— 1 df. (3.18) An easier way to establish the positive-definiteness of a^(u,v) is the following. For Ai,..., A n in C and for real numbers si,..., s n we write n 2 AfcV exp (— |s fc - s*|) k,e =i n = 2 AfcVmin (exp(-(s fc - s/)),exp(r i (s< - s k ))) k/= 1 96 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales = exp(—s fc )A fc exp(—s^A^min (exp(2s fe ),exp(2s^)) k,£= 1 f ex P ( 5 /c) / ^fcl[0,exp(2sjfc)] (0 k —1 2 <%> 0 . A similar argument can be used to prove that the function a Q (u,v) is positive dehnite. □ NORWEGIAN EFMD BUSINESS SCHOOL AffiroiTro Empowering People. Improving Business. Norwegian Business School is one of Europe's largest business schools welcoming more than 20,000 students. Our programmes provide a stimulating and multi-cultural learning environment with an international outlook ultimately providing students with professional skills to meet the increasing needs of businesses. B! offers four different two-yea i; full-time Master of Science (MSc) programmes that are taught entirely in English and have been designed to provide professional skills to meet the increasing need of businesses.The MSc programmes provide a stimulating and multi¬ cultural learning environment to give you the best platform to launch into your career * MSc in Business * MSc in Financial Economics * MSc in Strategic Marketing Management * MSc in Leadership and Organisational Psychology www.bi.edu/master Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales We now give existence theorems for the Wiener process (or Brownian motion), for Brownian bridge and for the oscillator process. 3.9. Theorem. The following assertions are true. (a) There exists a probability space (Cl, £F, P) together with a real-valued, Gaussian process {b(s) : s ^ 0}, called Wiener process or Brownian mo¬ tion, such that E (b(s)) = 0 and such that E (b(si)b(s 2 )) = min(si,s 2 ) for all s i, s 2 ^ 0. (b) Fix t > 0. There exists a probability space (Cl, T, P) together with a real¬ valued Gaussian process (X t (s) : t ^ s ^ 0}, called Brownian bridge, such that E(W(s)) = 0 and such that E(X t ( Sl )X t (s 2 )) = min(s 1 , s 2 ) - ^ for all s i, s 2 e [0, t]. (c) There exists a probability space (Cl, T, P) together with a real-valued Gaussian process { q(s ) : s e M}, called oscillator process, which is cen¬ tered, i.e. E (q(s)) = 0 and which is such that E (q(s 1 )q(s 2 )) = | exp (- |si - s 2 \) for all si, s 2 e M. (d) There exists a probability space (Cl, T, P) together with a real-valued Gaussian process (X(s) : s ^ 0}, called Ornstein-Uhlenbeck process, such that E(X(s)) = 0 and such that E(X(si)X(s 2 )) = ^ exp(—(si + s 2 )) (exp(2 min(si, s 2 )) - l) (3.19) = ^ (exp (- jsi - s 2 |) - exp(-(si + s 2 ))) for all si, s 2 ^ 0. 2. Brownian motion and related processes In what follows x and y are real numbers and so is p. Let {b(s) : s ^ 0} be Brownian motion (starting in 0) on a probability space (12, T, P) (i.e. E [b(s) ] = 0 and E [b (si) b (s 2 )] = min (si, s 2 )). Then the process {x + b(s) + ps : s ^ 0} is a Brownian motion with drift p starting at x. Let (X t (s) : 0 ^ s < t} be a Brownian bridge on a probability space (Q,T, P). Then the process s >—> ^1 — x + —y + -^t( s ) o < s ^ t is called pinned Brownian motion, namely pinned at x at time 0 and pinned at y at time t. Let {bj(s) : s ^ 0}, 1 < j G d, be d independent Brownian motions on the probability space (O. T, P). The process {(bi(s),..., ba(s)) : s ^ 0} is called d-dimensional Brownian motion. The characteristic function for d-dimensional Brownian motion starting at x e is given by: = i{#£ k ) min ( Sj ,s k ) (3.20) 98 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales where xq = x and where 0 = so < si < • • • < Sn- A similar definition can be given for d-dimensional Brownian bridge and for the d-dimensional oscillator process. Notice that a d-dimensional process {b(s) = (&i(s),..., bd(s)) : s ^ 0} is a d-dimensional Brownian motion, starting at 0, on the probability space (f2,T, P) if and only if E (bj(si), 6 fc(s 2 )) = min(si, s 2 ). Let us prove the above equalities. 3.10. Theorem. Let 0 = so < si < ■ • • < s n < co. Fix the vectors x and £i,..., £ n Put so = 0 and xq = x. The following equalities are valid: X ( 5 * “ s ^-i) 1=1 IN j=z X m(sj,s k )] j,k=1 n dx i... I dx n exp -i X (€j,Xj) JR d JR d \ j = 1 n n exp 1 X j x j-i\ 2 (sj — Sj_ij ?=i (V 2?r ( s i “ s i-i)Y = exp | -i ex P ^ X (O^fc)min (.s J; .sy) j . (3.21) (3.22) (3.23) For a and b e M. d we write (a + bi ) 2 = |a | 2 + 2 i (a, b) — \b\ 2 . Proof. In order to see the first equality we write X (se - s e ~ 1 ) 1=1 IN J=L i=i 3 l,32=Z n n min(ji ,j 2 ) = — s £-l) (Oi’^) = ( 5 min(ji,j 2 ) — 5 o) (£ 71 ? £ 72 ) iij2 = l ^=1 31,32=1 n n — ^ 5 min(ji,j 2 ) (Cn 5 £ 72 ) • = ^ min («Sji, «Sj 2 ) (^ji, ^ 2 ) • (3.24) Ji 5J2 — 1 31,32 = 1 For the second equality we proceed as follows: dx 1 ... dx n exp -i V (^, x?) \ jXi y n n 7=1 (V 2 tt (sj -Sj.i))' exp - |Xj — Xj_l| 2 (sj — Sj-ij (3.25) (substitute Xj = x + yi + • • • + yf) exp — 1 n \ r r ( n / n \ X x ) dyi...\ dy n exp -z X ( X £b dr jTi / V r=i \fr £ , 99 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales n 7 = 1 (V 2?r ( S i - S j-l)) (substitute yg = (s£ — sg-i ) 1 ' 2 zg) exp I Vj exp — i 2 ( s j s i-i) dz„ 1 Xi fe’ X ) dz !••• j =1 J JR d JR ex P [ -i 2 (si - Se- 1) 1/2 2 ] fl —= £=i j=£ / i=i (V2- exp = exp ( -i (&, a;) ) exp -- 2 («* - «*-i) 3 = 1 £=1 J=£ fl 7 7=^ f ^ exp ( ~ ( z e + i^/si - s t -i 2 & ) ]> £=i (v 27 t) y z \ j=£ ) J 100 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales From Cauchy’s theorem, it then follows that dxi ... I dx n exp -i V xj JR d jR d \ j =1 n n 7=1 (V2tt ( Sj ,))' exp Xj — Xj- 1 | 2 ( 5 j = exp n ^ j ex p 2 - s ^-i) J* ^ dze exp (-i |^| 2 ) j=e = 1 (V27T) exp Y j ex P 2 ( Si ~ ^-i) 2 & i=£ (first equality) = exp i This completes the proof of Theorem 3.10. 1 Y (c> x )j ex p Y &■> &> min Sfc )j (3.26) (3.27) (3.28) (3.29) (3.30) □ In the following proposition we collect a number of interesting properties of the (finite dimensional) joint distributions of some of the Gaussian processes we introduced so far. 3.11. Proposition. Let { b(s ) : s 5= 0} be d-dimensional Brownian motion and let OG(s) : 0 s t} be d-dimensional Brownian bridge. In addition let x and y be vectors in W 1 and let Q : M. d —» M. d be an orthogonal linear map. Also fix a strictly positive number a. (a) The joint distributions of the processes {b(s) : s > 0} and |s6 (i) : s > 0 coincide. (b) The joint distributions of the processes { b(as) : s ^ 0} and {\Iab(s ) : s ^ 0} coincide. (c) The joint distributions of the processes {q(s) : s e R} and <e s b ,2s : s e coincide. 101 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales (d) The joint distributions of the processes {X(t) : t ^ 0} and le t b f 2t - 1 t ^ 0 coincide. The process {X(t) : t ^ 0} also possesses the same joint dis¬ tribution as j^exp (—(t — s)) db(s ) : t ^ o|. (e) The joint distributions of the following processes also coincide: l — - ) x + -y + [I — -) b st t — s (i — f) x + fV + K s ) — o 0 < s < t, (3.31) 0 < s < t, (3.32) 0 < s < t. (3.33) (f) The process { Qb(s ) : s ^ 0} d-dimensional Brownian motion and so its joint distribution coincides with that of {b(s) : s ^ 0}. Notice that instead of the “distribution” of a random variable or a stochastic process, the name “law” is in vogue. 3 . 12 . Remark. Put b x (t ) = x + b(t). Then {b x (t) : t ^ 0} is Brownian motion that starts in x. PutX^t) = exp(— t)x + X(t). Then the process [X x (t] : t ^ 0} is the Ornstein-Uhlenbeck process of initial velocity x. 3.13. Remark. The stochastic integral $*exp(— (t — s))db(s) can be defined as the L 2 -limit of Xu=i ( b(sj ) — 6 (s,-_i)), whenever max (sj — tends l^j^n to zero. Here 0 = s 0 < Si < • • • < s n = t is a subdivision of the interval [0, t], 3.14. Remark. Let / : —*• C be a bounded Borel measurable function. Then E [f(X x (t))] is given by E [/ (X*(t))] = f / (e-’x + vT^F 5 !/) l dy , J ' 7 (Vtt) Moreover the Ornstein-Uhlenbeck process is a strong Markov process. 3.15. Remark. Let { b x (t ) : f ^ 0} be Brownian motion that starts at x (and has drift zero). Fix s > 0. The processes {b x (s + t) — b x (s) : t ^ 0 } and {b x (t) — x : t ^ 0 } possess the same (joint) distribution. In order to see this one may calculate the Fourier transforms, or characteristic functions, of their distributions. 3.16. Remark. Suppose that the Markov process {(O, T, P x ), (X(t),t > 0 ), OM ^ 0 ), (R", £)} (3.34) is Brownian motion in M n , and put Po(t, x. if) = - 7 - exp (2t vt) n/2 t > 0, x, y e R n . Define the measure jiff by ho V x ( A ) = E x [lAPo(t ~ s,X(s), y)] , (3.35) 102 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales where the event A belongs to T s = a (X(u) : u ^ s), for s < t. Since the process s i—> po(t — s,X(s),y) is a P x -martingale on the half-open interval 0 < s < t, it follows that the quantity is well-defined: its value does not depend on s, as long as A belongs to T s and s < t. From the monotone class theorem it follows that pgf can be considered as a positive measure on the cx-field given by T t _ = a (X(s) : 0 < s < t). Then the measure defined in (3.35) is called the conditional Brownian bridge measure. It can be normalized upon dividing it by the density po(t,x,y). PROOF of Proposition 3.11. Since all the indicated processes are d-dim¬ ensional Gaussian (the definition of a d-dimensional Gaussian process should be obvious: in fact in the discussion of 3.4 and in Theorem 3.7. The expected value p should be map from I to W l and the entries of the diffusion matrix a should be d x d-matrices), it suffices to show that the corresponding expectations and covariance matrices are the same for the indicated processes. In most cases this is a simple exercise. For example let us prove (f). Let q(k,£) be the entries of the matrix Q. Then d d E ((QKs 1 )) j 0 Qb(s 2 )) k S j = q(j,m) £ q(k,n)E(b m ( Sl )b n (s 2 )) m— 1 n=l d d d = 2 q(j,m) q(k,n)6 mtn mm(s 1 ,s 2 ) = £ q(j, m)q(k , m) min(si, s 2 ) 771= 1 77=1 777=1 = ( QQ *) (j, k) min(s!, s 2 ) = S jJc min(si, s 2 ). (3.36) This proves that {Qb(s) : s ^ 0} is again d-dimensional Brownian motion. This completes the brief outline of the proof of Proposition 3.11. □ 103 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In the proof of the existence of a continuous version of Brownian motion, we shall employ the following maximal inequality of Levy. 3.17. Theorem. (Levy) Let X 1: ..., X n be random variables with values in W 1 . Suppose that the joint distribution of X i,..., X n is invariant under any change where 6j = +1. Put Sk = < 2P(|S' n | Ss A). (3.37) < 2P (S n ^ A). (3.38) Proof. We prove (3.37). Put fc-i A k = f| {\Sj\ < A} n {\S k \ > A} 3 = 1 and put A = ULi A k- Write T k = X r Then ^ = \ S n + \? k and so {\S k \>\}<={\S n \>\}u{\T k \>\}. Hence, from the invariance of the joint distribution of (A 1? • • • • X n ) under sign changes we see P(A k )=P(A k ,\S k \>\) A IP (A k , |iSn| ^ A) + P {A k , \T k | ^ A) = 2P (A k , IS^I ^ A). Since the events A k , 1 ^ k ^ n, are mutually disjoint, we infer P ( max I S k I ^ A n n = P(A) = P (A k ) < 2 P (A k , |S n | > A) < 2P (\S n \ > A). k =1 1 This proves (3.37). The proof of (3.38) is similar and will be left to the reader. Altogether this completes the proof Theorem 3.17. □ Let {X(f) : t ^ 0} be Brownian motion on the probability space (12,T, P). We shall prove that there exists a continuous process {6(f) : f ^ 0}, that is indistin¬ guishable from the process {X(t) : t ^ 0}. This means that P(A(f) = 6(f)) = 1 for all t ^ 0. 3.18. Theorem. Let {X(t) : t ^ 0} be Brownian motion on some probability space (12, T, P). Then there exists stochastic process {6(f) : f ^ 0} which is P- almost surely continuous, and that is also a Brownian motion on the probability space (12, T, P) and that is indistinguishable from the process {X(s) : s ^ 0}. Here we suppose that T contains the P -zero sets. of sign (xi,...,x n ) faxi ,..., e n x n ), Then for any A > 0 If d = 1, then P ( max I S k \ A A ) P ( max S k ^ A V l^k^n 104 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Proof. Without loss of generality we may and do assume that the Brownian motion {X(s) : s ^ 0} has drift 0 and diffusion matrix identity. For the proof we shall rely on Theorem 3.17 and on the Borel-Cantelli lemma, which reads as follows. Let (A n : n e N) be a sequence of events with Xm=i P(Ai) < 00 • Then p (rCi ix , m A n ) = 0. In Theorem 3.18 we choose the sequence ( A n :neN) as follows. Let D be the set of non-negative dyadic rational numbers and put I A„ = < max sup \X(q)-X(k2- n )\>-[. I 0^fc<n2" ?eDn [ fe2 -n ( fe+1 ) 2 -n] An application of Theorem 3.17, with X(t+j82~ m ) — X(t+(j — l)82~ m ) replacing Xj yields P ( rnax m \X (t + jS2~ m ) - X(t) \ > a) < 2P(| X(t + 8) - X(t) \ ^ a) L exp H < ^jE\X(t + 5) -X(t)\ 4 = 26 1 « 4 (V27T)" '2 \y\ ) \y\ d v 28 2 (2d + d 2 ) cr In (3.39) we let m tend to infinity to obtain: sup | X(t + qS) — X(t)\ > a J < 0^q^l,qeD 28 2 (2d + d 2 ) a q (3.39) (3.40) Hence, with J n ^ = [k2 n , (k + 1)2 n ] (see also (3.46) below), and with t = k2 n and 8 = 2~ n , max sup \X(q) — X(k2 n )| > — o^k<n 2 n qeDn Jn k n < n2 n —1 2 ' k =0 sup \X(q)-X(k2~ n )\>- qeDnJ ntk n < n2' , 2 (2d2~ 2n + d 2 2 —2n\ 2 (2d + d 2 )n 5 6d 2 n 5 (3.41) Since the sequence in (3.41) is summable, we may apply Borel-Cantelli’s lemma to conclude that P-almost surely, for all t > 0, the path q >-» X(q) is uniformly continuous on D n [0, t] . So it makes sense to define the P-almost surely continu¬ ous function s i—> b(s) by b(s) = lim ? ^ Sj96 D X(s). It is not so difficult to see that the process {b(s) : s 5= 0} is also a Brownian motion. In fact let £i,..., be n vectors in M, d and suppose 0 = so < si < • • • < s n . Then we choose sequences 0 = q 0 (m) < si < qi(m) < s 2 < <? 2 (^) < s n -1 < < q n -i(m) < s n < q n (m ), m e N, in D, such that qk(rn) [ Sk, if rn tends to infinity and this for 1 < k < n. Since {A"(s) : s A 0} is d -dimensional Brownian motion we have E exp (&,X(q k (m))) k =1 105 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales = exp | — - > , 2 k—j In (3.42) we let m tend to go to obtain 1 n 3 = 1 - qj-i(m)) . (3.42) E ^exp (tk,b(s k ))^j j =exp(-ij] k—j (sj-Si-i) ■ (3.43) This equality shows that {b(s) : s ^ 0} is a Brownian motion. In order to prove that it cannot be distinguished from the process (X(s) : s 5= 0}, we notice first that E (exp (-* <£, X(t + s)~ X(t)))) = exp (-g |£| 2 sj , ^ E d . (3.44) Hence, for (el' 1 , E |exp (-i (£,X(t))) - exp (-* (f, b(t)))\ 2 = E(2 - exp (* (£,X(t) - b(t ))) - exp (—i (£,X(t) - b(t )))) = hm (2-E(exp (~i(£,X(q) — X(t)))) — E (exp (—i (£,X(t) — X(q))))) qit,qeD = 2 — 2 hm exp ( — |£| 2 (q»-1) ) = 0. qlt,qeD V 2 V ’ J (3.45) From (3.45) it readily follows that the processes {X(s) :s ^ 0} and {b(s) :s ^ 0} cannot be distinguished. □ Brain power By 2020, wind could provide one-tenth of our planet’s electricity needs. Already today, SKF’s innovative know¬ how is crucial to running a large proportion of the world’s wind turbines. Up to 25 7o of the generating costs relate to mainte¬ nance. These can be reduced dramatically thanks to our (^sterns for on-line condition monitoring and automatic lul|kation. We help make it more economical to create cleanSkdneaper energy out of thin air. By shsfefig our experience, expertise, and creativity, industries can boost performance beyond expectations. Therefore we need the best employees who can kneet this challenge! Power of Knowledge Engineering Plug into The Power of Knowledge Engineering. Visit us at www.skf.com/knowledge 106 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In this proof of Theorem 3.18 we have also used the fourth moment E|X(t + s) -X(£)| 4 . From (3.44) it follows that this moment does not depend on t and hence E | X(t + s) - X(t) | 4 = E | X(s) - X(0)| 4 = E |X(s)| 4 . A way of computing E |X(s)| 4 is the following: E |X( S )| 4 = E (exp (-i «, AT(«)») | £ . 0 = (2 ds 2 - 2s 3 |£| 2 + s 4 |£| 4 - 2ds 3 |£| 2 + d 2 s 2 ) exp (-1 |£| 2 ) | ?=0 = (2 ds 2 - 2 (d + 1 )s 2 |£| 2 + s 4 |£| 4 + d 2 s 2 ) exp (-1 |£| 2 s) | {=0 = 2 ds 2 + d 2 s 2 . (3.46) In the following theorem we compute the finite dimensional distributions of d- dimensional Brownian motion starting at 0 and possessing drift fi. Therefore we define the Gaussian kernel p{t ) x, y ) by p(t, x, y) = -— exp (2tt/ ) Notice the Chapman-Kolmogorov identity p(s, x, z)p(t , y) = p(s + t, x, y)p st s t sx + ty \ s + t ,Z ) ‘ 3.19. Theorem. Let { b(s ) : s ^ 0} be d-dimensional Brownian motion with dif¬ fusion matrix identity, with drift 0 and which starts in 0. Let f n be bounded Borel measurable functions on R. d and let 0 = s 0 < Si < • • • < s n . Then ^ + K s j ) + l LS j) 0=1 r r n n ... dx 1 ...dx n y\f j (x j )Y\p(& JR d JR d + 1 (3.47) Sj —15 X j —1 llSj- U Xj /iSj) , where x 0 = x. 3.20. Remark. Equality (3.47) determines the joint distribution of the process |^A(s) := x T 6(s) T ys \ s ^ 0}. This will follow from the monotone class theorem. The vector y is the so-called drift vector and the process X = {X(s) : s ^ 0} starts at x in 107 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.21. Remark. Another consequence of equality (3.47) is the fact that the random vector b(t) — b(s), t > s fixed, is independent of the cr-field generated by the process {b(a) : 0 < a ^ s}. This fact also follows from (3.48) below together with the monotone class theorem. For £ e M d , e M d , 1 < j < n, t > s ^ s n > ■■■ si > «o = 0 the following identity is valid and relevant: E ( exp ( -i (f, bit) - b(s)) - i ^ (&, K s j)) 3 = 1 ex P (l^l 2 (t - s) - \ 2 min ( S i> S fc) <&>&) j,k=l = E (exp (-* (f, b(t) - b(s)))) E ( exp ( -i J] 6(sj-)) i=i (3.48) In other words a Brownian motion (diffusion matrix identity) is a Gaussian process {b(s) : s ^ 0} with independent increments b(t) — b(s), t > s, with mean p(t — s) and covariance matrix co v(bk(t) —bk(s), be(t) —be(s)) = 5k/(t — s). PROOF. Theorem 3.10 shows that the equality in (3.47) holds for functions fj, 1 < j < n, of the form fj( x ) = J exp (-*(£, z))d/Ji(0» (3-49) where fij = 6^ is the Dirac measure Fubini’s theorem then implies that (3.47) also holds for functions fj, 1 A j A n, of the form (3.49) with l-ij(B') = w ith gj e L l (R d ), K jT n. Since, by the Stone-Weierstrass the¬ orem functions of the form (3.49) with /q ( B ) = <p(£) df where gj e L 1 (PA), are dense in the space Co (R d ), it follows that (3.47) holds for functions fj e Co ( M . d ), 1 A j A d. By approximating indicator functions of open subsets from below by functions in Co (PA) it follows that the equality in (3.47) holds for functions fj which are indicator functions of open subsets. A Dynkin argument (or the monotone class theorem) then shows that (3.47) is also true if the func¬ tions fj are indicator functions of Borel subsets Bj, 1 < j < n. But then this equality also holds for bounded Borel functions fj, 1 A j A n. This completes the proof of Theorem 3.19. □ Next we want to define standard Brownian motion, with drift vector p, that starts at rel* 1 . 3.22. Definition. The standard Brownian motion, starting at x e R. d and with drift p is defined as the canonical Gaussian process {A(.s) : s P 0} defined on (D, T, P x ) with the property that the increments X(t + h) — X(t) are mutually independent and have P x -expectation ph. Moreover it starts P x -almost surely at x, i.e. P x (A(0) = x) = 1 and co <v (Xk(t + h) — Xk(t),Xg(t + h) — Xg{t + h)) = 6k,eh. The covariance is of course also taken with respect to P x . The process is canonical because for fl we take fl = C ([0, co), W l ) , for X(t) we take X(t)(u) = 108 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales cu(£), uo e Q. For £F we take the cr-field in generated by the state variables {X(s) : s ^ 0}. For all this we often write {(ft, T, P x ), (X(t) (R d , 3 )} . Here the shift or translation operators $ t , t F 0, are defined by i) t (oj)(s) = u>(s + t), uj e ft. We also introduce the filtration (SF t : t F 0) defined as the full history: “J t is the cr-field generated by the variables X(s), 0 < s < t. We also shall need the right closure 3y + defined by 3y + = n s>t ?s. In the following result we give some interesting martingale properties for Brow¬ nian motion. 3.23. Proposition. Let {(ft, T, P x ), (X(t) : t> 0), (d t : t> 0), (M d , 3 )} be standard Brownian motion that starts at x e R d and that has drift p. For t > s the variable X (t) — X(s) does not depend on the o-field T s . The following processes are F x -martingales with respect to the filtration T t , t 5= 0: 1 > X(t) — tp, t i—> |X(f) — tp | 2 — dt. Proof. The fact that the increment X(t) — X(s) does not depend on the past T s is explained in Remark 3.21 following Theorem 3.19. The other asser¬ tions are consequences of this. Let s and t be positive real numbers. Then we have (X(s + t) — (s + t)p | T s ) — (X(s) — spi) = E x (X(s + t)-X(s) | T s )-tp (increments are independent of the past) = E x (W(s + t) — X(s)) — tp = tp — tp = 0. (3.50) Similarly, but more complicated, we also see Ex [| X{s + t)-(s + t)p\ 2 - d(s + t)~ |X(s) - sp\ 2 + ds | T s ] = E x [|X(s + t) — X(s) — tp + X(s) — sp\ 2 — dt — |X(s) — sp\ 2 | T s ] = Ex [|X(s + t) — X(s) — tp\ 2 — dt + (X(s + t) — X(s) — tp,X(s ) — sp) | T s ] (use (3.50)) = Ex [|X(s + t) — X(s) — tp\ 2 | T s ] — dt (again an application of (3.50)) = Ex [|X(s + t)~ X(s) - tp\ 2 ] - dt d = ^ cov (Xk(s + t) — Xk(s), Xk(s + t) — Xfc(s)) — dt = dt — dt = 0. (3.51) k =1 This proves Proposition 3.23. □ 109 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales So far we have looked at Brownian motion as a Gaussian process. On the other hand it is also a Markov process. We would like to discuss that now. In fact mathematically speaking equality (3.47) in Theorem 3.19 is an equivalent form of the Markov property. As already indicated in Remark 3.20 following Theorem 3.19 the monotone class theorem is important for the proofs of the several versions of the Markov property. 3.24. Definition. Let Cl be a set and let S be a collection of subsets of Cl. Then S is a Dynkin system if it has the following properties: (a) 9e§; (b) if A and B belong to S and if A 3 B, then A\B belongs to S; (c) if (A n : n e N) is an increasing sequence of elements of S, then the union Un= i A* belongs to S. Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales The following result on Dynkin systems is well-known. 3.25. Theorem. Let M be a collection of subsets of Cl, which is stable under finite intersections. The Dynkin system generated by M coincides with the a- field generated by M. 3.26. Theorem. Let Cl be a set and let M be a collection of subsets of Cl, which is stable (or closed) under finite intersections. Let J~C be a vector space of real valued functions on Cl satisfying: (i) The constant function 1 belongs to TC and 1 a belongs to TC for all A e M; (ii) if (f n : n e N) is an increasing sequence of non-negative functions in TC such that f = sup neN f n is finite (bounded), then f belongs to TC. Then TC contains all real valued functions (bounded) functions on Cl, that are cr(M) measurable. Proof. Put D = [A ^ Cl : 1 a e IT}- Then by (i) Cl belongs to D and D ^ M. If A and B are in D and if B ^ A, then B\A belongs to D. If ( A n : n e N) is an increasing sequence in D, then l U A n = sup n 1 A n belongs to T) by (ii). Hence D is a Dynkin system, that contains M. Since M is closed under finite intersection, it follows by Theorem 3.25 that D 3 <r(M). If / > 0 is measurable with respect to <r(M), then / = sup2- n ^! 1 ( 3 - 52 ) n J 1 Since 1 {f^j 2 ~ n }, j , n e N, belong to we see that / belongs to ! Ji. Here we employed the fact that cr(M) c X>. If / is a(M)-measurable, then we write / as a difference of two non-negative a( M)-measurablc functions. □ The previous theorems (Theorems 3.25 and 3.26) are used in the following form. Let Cl be a set and let ( E t , Lf) ieI be a family of measurable spaces, indexed by an arbitrary set /. For each i e I, let S. ( denote a collection of subsets of E,, closed under finite intersection, which generates the cr-field £ ( , and let f, : Cl —* E, be a map from Cl to E{. In this context the following two propositions follow. 3.27. Proposition. Let M be the collection of all sets of the form fi 1 (A) ) Ai e ieJ i e J, J c / 1 J finite. Then M is a collection of subsets of Cl which is stable under finite intersection and cr(M) = a (fi : i e I). 3.28. Proposition. Let TC be a vector space of real-valued functions on Cl such that: (i) the constant function 1 belongs to J~C; (ii) if ( h n : n e N) is an increasing sequence of non-negative functions in TC such that h = sup n h n is finite (bounded), then h belongs to J~C; (iii) TC contains all products of the form Idiej 1 a, 0 fi, J *= I, J finite, and Aj e §i, i e J . ill Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Under these assumptions TC contains all real-valued functions (bounded) func¬ tions in cr(fi : i e I). The Theorems 3.25 and 3.26 and the Propositions 3.27 and 3.28 are called the monotone class theorem. In the following theorem T is the cx-field generated by {X(s) : s A 0} and % is the cr-field generated by the past or full history, i.e. = cr (X(s) : 0 A s T t\. If T is an (9y+)-stopping time we write J T+ = f){AeT:An{T^t}e % + }. tjs o An (3y+)-stopping is an T-measurable map T from Q to [0, co] with the property that {T < t} belongs to 3 r t + for all t ^ 0. Notice that stopping times may take infinite values. Often this is very interest¬ ing. 3.29. Theorem. Let {(f2,T, P*) , (X(t) : t ^ 0), (& t : t ^ 0), (E d ,!B)}, x e R d , be d-dimensional Brownian motions. Then the following conditions are verified: (ai) For every a > 0, for every t ^ 0 and for every open subset U ofM. d , the set {jeRh P x (X(f) 6 U) > a) is open; (a 2 ) For every a > 0, for every t ^ 0 and for every compact subset K of the set {x e : P x (X(t) e K) ^ aj is compact; (b) For every open subset U of R d and for every x e U, the equality limP x (X(t) e U) = 1 is valid. Moreover d-dimensional Brownian motion has the following properties: (i) For all t ^ 0 and for all bounded random variables Y : Ll —► C the equality E x (Totf t |T t )=E x( p(T) (3.53) holds P x -almost surely for all x e M d ; (ii) For all finite tuples 0 < t\ < t 2 < ... < t n < go together with Borel subsets B\,..., B n ofR d the equality IPa; (Al(tl) £ B\,... ,X(t n ) e B n ) I • • • I I P(fn tn— 1 ; •^n—li dx r (jP(t n _\ t n _ 2 , X n —2i dx n —fi J B± J B n —i J B n ... P(t ,2 — dx 2 )P(t u x, dx i) (3.54) is valid for all x e (here P x (A(f) e B) = P(t, x, B)); (in) For every (£F t+ )-stopping time T and for every bounded random variable Y : fl —> C the equality E x (Yod T \ T t+ ) = E X (T) 00, (3.55) holds P x -almost surely on {T < oo} for all x e R d ; 112 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales (iv) Let ¥>i be the Borel field of [0, go). For every bounded function F : [0, go) x Cl —> C, which is measurable with respect to and for every (%+)-stopping time T the equality E x ({u> ~ F(T(u),Mu))} I ^t+) = {cS ~ E* (IV)) W - F(T(u'),u)}} (3.56) holds P x -almost surely {T < go} for all leKf Since (/-dimensional Brownian motion verifies (ai), (a 2 ) and (b), the properties in (i), (ii), (iii) and (iv) are all equivalent. Properties (i) and (ii) are always equivalent and also (iii) and (iv). The implication (iii) => (ii) is also clear. For the reverse implication the full strength of (ai), (a 2 ) and (b) is employed. The fact that Brownian motion possesses property (ii) is a consequence of Theorem 3.19. In fact the right continuity of paths is very important. Since we have proved that Brownian motion possesses continuous paths P x -almost surely this condition is verified. Property (i) is called the Markov property and property (iii) is called the strong Markov property. Equality (3.56) is called the strong time-dependent Markov property. We shall not prove this result. It is part of the general theory of Markov processes and their sample path properties. It is also closely connected to the theory of Feller semigroups. As in Theorem 3.29 let {(Cl, £T, P x ) ,(X(t) : t ^ 0), (i?i : t ^ 0), (R d , B)} be Brownian motion starting in x. In fact the family of operators {P(t) : t ^ 0} defined by [P(t)f] (x) = (f(X(t)), f e L co (R d ), t ^ 0, is a Feller semigroup, because it possesses the properties mentioned in the following definition. In what follows E is a second countable locally compact Hausdorff space, e.g. E = M'b We define a Feller semigroup as follows. 3.30. Definition. A family {P(t) : t ^ 0} of operators defined on L co (E ) is a Feller semigroup, or, more precisely, a Feller-Dynkin semigroup on Cq(E) if it possesses the following properties: (i) It leaves Cq(E) invariant: P(t)Co(E ) c Cq(E) for t ^ 0; (ii) It is a semigroup: P(s + 1) = P(s) o P(t) for all s, t ^ 0, and P( 0) = /; (iii) It consists of contraction operators: ||T > (i)/|| 00 A ||/’|| 00 for alH 5= 0 and for all / e C'o(A); (iv) It is positivity preserving: / ^ 0, / 6 Cq(E), implies P(t)f ^ 0; (v) It is continuous for t = 0: lim t | 0 [P(t)f] ( x ) = f( x ), for all / e Cq(E) and for all x e E. In the presence of (iii) and (ii), property (v) is equivalent to: (v') lim ti o | \P(t)f ~ fL = 0 for all / e C 0 (E). So that a Feller semigroup is in fact strongly continuous in the sense that, for every / e C 0 (E), lim \\P(s)f - P(t)f\\ x = 0. S^-t 113 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales It is perhaps useful to observe that Co(E), equipped with the supremum-norm ||• Ioq is a Banach space (in fact it is a Banach algebra). A function / : E —* C belongs to Cq(E) if it is continuous and if for every e > 0, there exists a compact subset K of E such that \f(x)\ < e for x $ K. We need one more definition. Let {P(t) : W 0} be a Feller semigroup. Define for U an open subset of E, the transition probability P(t , x, U), t ^ 0, x e E, by P(t, x, U ) = sup {[P(t)u] (x) : 0 ^ u < 1[/, u e Cq(E)} . This transition function can be extended to all Borel subsets by writing P(t,x,K) = utf{P(t,x,U):U open U 3 R ), for K a compact subset of E. If D is a Borel subset of E, then we write P(t, x, B ) = inf {P(t, x,U) : U 3 B,U open } = sup {P(t, x,K) : K ^ B,K compact } . It then follows that the mapping B P(t,x,B ) is a Borel measure on £, the Borel field of E. The Feller semigroup is said to be conservative if, for all t ^ 0 and for all x e E, P(t, x, E) = 1. qaiteye Challenge the way we run EXPERIENCE THE POWER OF FULL ENGAGEMENT... RUN FASTER. — r RUN LONGER.. RUN EASIER... > ww • 114 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales We want to conclude this section with a convergence result for Gaussian pro¬ cesses. 3.31. Proposition. Let : s e I^J, n e N, be a sequence of Gaussian pro¬ cesses. Let (X s : s e I) be a process with property that E [X U X V ] = lim E [V re) V re) l, for all u and v in I and E [X u ] = lim E [ V n) l, for all uel. n —>00 L J Also suppose that, in weak sense, for all finite subsets (ui,..., u m ) of I. Then the process (X s : s e I) is Gaussian as well. Proof. Let £i,...,be real numbers, and let ui,... ,u m be members of I. Then E ^exp ^ f k X^> / m -t ui = exp -i 2 &E (X£>) - - 2 &&eov (*£>,*«) k =1 k,e =i Next let n tend to infinity to obtain (here we employ Levy’s theorem on weak convergence): E ( exp ( -i ^ £kX Uk k =1 m = exp f -i 2 &E [X Uk ] - - 2 &&E [(X ut - E (X„J) (X u , - E (A'„,))] \ . I k=l Z M=i So the result in Proposition 3.31 follows. □ For Levy’s weak convergence theorem see Theorem 5.42. 3.32. Theorem. Brownian motion is a Markov process. More precisely, (3.53) is satisfied. Proof. Let F be a bounded stochastic variable. We have to show the following identity: E x [F o3 t | 3y] = E X (t) [F] , P^-almost surely. It suffices to show that E* [F o3 t x G] = E x [E x(t) [F] G] (3.57) for all bounded stochastic variables F and for all bounded dVmeasurable func¬ tions G. By an application of the monotone class theorem twice (see Proposi¬ tion 3.28) it suffices to take F of the form F = Iljli fj (X ( Sj )), 0 < Si < s 2 < 115 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales • • • s m < oo, and G of the form G = =1 9j (X (; tj )), 0 < ti < t 2 < ■ • • < t n < t. Here fi,... ,f m and gi , • • •, g m are bounded continuous functions from M, d to R or C. Once the monotone class theorem is applied to the vector space {G e L 00 (0, 5 t ) : E x [E x(t) [F] x G\ = E* [F o x G]} , where F is as above, and once to the vector space {F e L 00 (Q, T) : E X (t) [F] =E x [Foi3 t | J t ] , P x -almost surely} . Then (3.57) may be rewritten as E* [h {X ( Sl + t)) • • • f m (X (s m + t)) g i (X (G)) ■■■g n (X (f n ))] = E x [E x(t) [h (X ( 5l )) • • • f m (X (s m ))] c h (X (G)) ■■■g n (X (*„))] . (3.58) Put Tj = tj, 1 GjGn, r n+k = s k + t, 1 < k G m; hj = 9j , HjX n, h n+k = f k , 1 < k < m. By definition we have E* [fi (X ( Sl (X ( Sm + t)) 9l (X (H)) (X (4))] = [A? (Tl)) ' * ‘ ^n+ra (X (^”n+m))] h-n+m (■T-n+m) P (^1) ’"T) ■Tl) ‘ P ('T’n+m '^n+m— 1> •T-n+m—1) %n+m) • (3.59) Next we rewrite the right-hand side of (3.58): E* [E x(t) [/! (X ( Sl )) • • • f m (X (s m ))] .91 (X (ti)) • • • g n (X (* n ))] = hi (X (ti)) • • • hn (X (£„)) J . . . J efi/i . . . d'ljmf l (hi) • • • fm (Urn) P 1 : X (t) , Hi) ’ ' ' P (^m 1 5 Um—l 1 Urn) • • • dz n gi (^1) • • • g n {%n) p (ii , x, Zi ) • • • p ( t n t n —\, , 2^) dzp (i z) --dy. „/,(»:)• ••/„(!/„) p(si,z,yi) ■■■p(s m - s m -i,hm-i,hm) (Chapman-Kolmogorov: ^p(t — t n , z n , z) p (si, z,y) dz = p(si + t — t n , z n ,y)) j ' (2l) (</.)• p (tl, X, Zi) • • • , z n ) P (^1 T t 2/l) ’ ‘ ’ P (^m 1 5 Um—l •> Vm) = E x [A (X (si + t)) • • • fm (X (s m + A) 91 (X (A)) ---g n {x ( t n ))]. (3.60) Since the expressions in (3.59) and (3.60) are the same, this proves the Markov property of Brownian motion. The proof of Theorem 3.32 is now complete. □ 116 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3. Some results on Markov processes, on Feller semigroups and on the martingale problem Let E be a second countable locally compact Hausdorff space, let E A be its one-point compactification or, if E is compact, let A be an isolated point of E A = E (J A. Define the path space 0 as follows. The path space 0 is a subset of with the following properties: (i) If c o belongs to Cl, if t ^ 0 is such that u){t ) = A and if s ^ t, then u>(s) = A; (ii) Put CM = inf {s > 0 : uj(s) = A} for uj e Cl. If cj belongs to Cl, then uj possesses left limits in E A on the interval [0, C] and it is right-continuous on [0, oo); (iii) If uj belongs to O, if t ^ 0 is such that uj(t) belongs to E, then the closure of the set {uj(s) : 0 < s ^ t} is a compact subset of E or, equivalently, if t > 0 is such that u ;(i—) = A and if s ^ t, then u ;(s) = A. 3.33. Definition. The random variable C, defined in (iii.) is called the life time of uj. A path uj e Cl is said to be cadlag on its life time. We also define the state variables X(t) : 0 —► E A by X(t)(uj) = X(t,u) = uo(t), t ^ 0, uo e Cl. The translation or shift operators are defined in the following way: [$ t (u;)](s) = uj(s + t), s, t > 0 and uj e Cl. The largest subset of (£ ,A )'-°’ 00 ^ with the properties (i), (ii) and (iii) is sometimes written as D ([0, oo), £’ A ) or as D e a ([0, oo)). Let T be a cr-field on Cl. A function Y : Cl —> C is called a random variable if it is measurable with respect to T. Of course C is supplied with its Borel field. The so-called state space E is also equipped with its Borel field £ and E A is also equipped with its Borel field £ A . The path uj/\ is given by wa(s) = A, s ^ 0. Unless specified otherwise we write O = D([0,oo) ,E A ). The space D([0, oo), E A ) is also called Skorohod space. In addition let T be a cr-field on Cl and let {3q : t ^ 0} be a filtration on Cl. Suppose £F t c T, t ^ 0, and suppose that every state variable X(t), t ^ 0, is measurable with respect to “Jf (This is the case where e.g. is the cr-field generated by (AT(s) : s < t}.) 117 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales We also want to make a digression to operator theory. Let L be a linear operator with domain D(L ) and range R(L ) contained in C${E). The operator L is said to be closable if the closure of its graph is again the graph of an operator. Here the graph of L, G(L), is defined by G(L) = {(/, Lf) : f e D(L)}. Its closure is the closure of G(L) in the cartesian product Cq{E ) x Co(E). If the closure of G(L) is the graph of an operator, then this operator is, by definition, the closure of L. It is written as L. Sometimes L is called the smallest closure of L. 3.34. Definition. Let L be a linear operator with domain and range in Cq(E). (i) The operator L is said to be dissipative if, for all A > 0 and for all /£ £>(£), II A/ - L/t A ||/L . (3.61) (ii) The operator L is said to verify the maximum principle if for every / e D(L) with sup {Re f(x) : x e E) strictly positive, there exists x 0 e E with the property that Re f{x o) = sup {Re fix) : x e E) and Re Lf(x o) < 0. (iii) The martingale problem is said to be uniquely solvable, or well-posed, for the operator L, if for every x e E there exists a unique probability P = Pa, which satisfies: (a) For every / 6 D(L) the process }(X(t)) - f(X( 0)) - f Lf(X(s))ds y t S 0, Jo is a P-martingale; (b) P(X(0) =x) = l. (iv) The operator L is said to solve the martingale problem maximally if for L the martingale problem is uniquely solvable and if its closure L is maximal for this property. This means that, if L\ is any linear operator with domain and range in Cq(E), which extends L and for which the martingale problem is uniquely solvable, then L coincides with L\. (v) The operator L is said to be the (infinitesimal) generator of a Feller semigroup {P(t ) : t> 0 }, if L s-lim t[ 0 m -1 t This means that a function / belongs to D(L ) whenever Lf := lim 40 mi - / t exists in Cq{E). An operator which verifies the maximum principle is dissipative (see e.g. [ 141 ], p. 14) and can be considered as kind of a generalized second order derivative operator. A prototype of such an operator is the Laplace operator. An operator for which the martingale problem is uniquely solvable is closable. This follows from (3.125) below. Our main result says that linear operators in Cq{E) which maximally solve the martingale problem are generators of Feller semigroups and conversely. 118 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.35. Definition. Next suppose that, for every x e E, a probability measure P,„ on T is given. Suppose that for every bounded random variable Y : 0 —*• M the equality E x (Y o f} t \ T t ) = Ex(p(T) holds P,-almost surely for all x e E and for allt ^ 0. Then the process m % P»), (X(t) :t>O),(0 t :t>O),{E,e.)} is called a Markov process. If the fixed time t may be replaced with a stopping time T, the process {(f2, T, P x ), (X(t) : t ^ 0), (# t : t ^ 0), (.E , £)} is called a strong Markov process. By definition Pa (A) = 1a(<^a) = d WA (d). Here A belongs to T. If the process {(fi, T, P^) ,(X(t) : t ^ 0), (# t : f ^ 0), (A, £)} is a Markov process, then we write P(f, x, .B) = F x (X(t) e B), t ^ 0, Be£, xeE, (3.62) for the corresponding transition function. The operator family {P(t) : t ^ 0} is defined by [P(t)f](x) = E x (f(X(t))), f £ C 0 {E). An relevant book on Markov processes is Ethier and Kurtz [54]. An elementary theory of diffusions is given in Durrett [45]. In this aspect the books of Stroock and Varadhan [133], Stroock [132] [131], and Ikeda and Watanabe [61] are of interest as well. We shall mainly be interested in the case that the function P(t)f is a mem¬ ber of Cq(E) whenever / is so. In the following theorem T is the cr-field generated by |A(s) : s ^ 0} and ‘J t is the a-field generated by the past or full history, i.e. T t = <t{X(s) : 0 < s ^ t}. If T is a stopping time we write T r+ = fU {A e T : A n {T < t} e T t+ }. 3.36. Theorem. Let (0,T, P x ), x e E, be probability spaces with the following properties: (ai) For every a > 0, for every t ^ 0 and for every open subset U of E, the set {x e E : P x {X{t) e U) > a} is open; (a 2 ) For every a > 0, for every t ^ 0 and for every compact subset K of E, the set {x e E : P x (X(t) e K) ^ a} is compact; (c) For every open subset U of E and for every x e U, the equality limP x (X(t) £ U) = 1 is valid. H o The following assertions are equivalent: (i) For all t ^ 0 and for all bounded random variables Y : Cl —> C the equality E x (Yof> t \? t ) = E x{t) (Y) (3.63) holds Pa ,-almost surely for all x e E; (ii) For all finite tuples 0 < t\ < f 2 < • • • < t n < oo together with Borel subsets Bi,... ,B n of E the equality Pr (X(ti) e B\,... ,X(t n ) e B n ) I • • • I I P(fn I'n— 1) T n _ i, C?X n )T > (t n _i t n —2i •^n— 2i dx n —f) JBi JB n -i JB n 119 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales ... Pfe — ti, Xi, dx 2 )P(ti, x, dx i) (3.64) is valid for all x e E (here ¥ x (X(t) e B) = P(t, x, B)); (in) For every (3y+) -stopping time T and for every bounded random variable Y : Cl —> C the equality E* (Y o d T | J T+ ) = E x(r) (F), (3.65) holds P x -almost surely on {T < oo} for all x e E; (iv) Let ¥> be the Borel field of [0, oo). For every bounded function F : [0, go) x Cl —> C, which is measurable with respect to 23®3q and for every (3^+) -stopping time T the equality E x ({a; > F (T(u), 'dr(^))} I 3 r r+) = {&' ► E X (T(ty)) {w *->■ F (T(u'), a>)}} (3.66) holds P x -almost surely {T < go} for all x e E. Equality (3.66) is called the strong time-dependent Markov property. We shall not prove this result. This e-book is made with SetaPDF QO SETASIGN PDF components for PHP developers www.setasign.com 120 Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.37. Theorem. Let {P(t) : t ^ 0} be a Feller semigroup. There exists a col¬ lection of probabilities (P x ) x6£; on ^ e cr-field T generated by the state variables {X(t) : t ^ 0} defined on Q := D ([0, oo) , 17 A ) in such a way that E x [f (X(h ),..., X(t n ))] = J / (Xfa),X(t n )) dP x ^ f (^U 1 j • • • > •En)P(fin In— 1 > %n—l> dx r fjP(t n —\ ^n—2) •^n—2i dx n — i) ... P (^2 ~ ti,Xi,dx2)P(ti,x, dx i), (3.67) where f is any bounded complex or non-negative Borel measurable function de¬ fined on E A x ... x 17 A ; that vanishes outside of E x ... x E. Let the measure spaces (Q,tP,¥ x ) xeE be as in (3.67). The process {(fi, ^ P*), (X(t) :t>O),{0 t :t>O),{E,e.)} is a strong Markov process. The proof of this result is quite technical. The first part follows from a well- known theorem of Kolmogorov on projective systems of measures: see Theorems 1.14, 3.1, 5.81. In the second part we must show that the indicated path space has full measure, so that no information is lost. Proofs are omitted. They can be found in for example Blumenthal and Getoor [ 20 ], Theorem 9.4. p. 46. For a discussion in the context of Polish spaces see, e.g ., Sharpe [ 120 ] or Van Casteren [ 146 ]. For the convenience of the reader we include an outline of the proof of Theorem 3.37. The following lemma is needed in the proof. 3.38. Lemma. Let (£4,35, P) be a probability space, and let t i—> Y(t), be a supermartingale, which attains positive values. Fix t > 0 and let D be a dense countable subset of [ 0, oo). Then Y(t)> 0 inf Y(s) = 0 <s<t, seD = 0 . (3.68) PROOF. Let (s_,-). be an enumeration of the set D n [0,oo). Fix n, N e N, and define the stopping time S nj jv by S u n = min \ s, : min Y (sA < 2~ n > . ’ ( IsSjsSV J ’ J Then we have Y (S n ^) < 2~ n on the event {S n> N < oo}. In addition, by the supermartingale property (for discrete stopping times) we infer E Y(t), min Y (sA <2 n \Y(t),S n , N <t] = E [Y (■ t ), min (S n , N , t) < t] < E [Y (min (S n>N , t)) , min (S n>N , t ) < t] = E [Y (S^n) , Sn t N < t] V E [2~ n , S n , N < t] ^ 2~ n . (3.69) 121 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Y(t), inf Y(s)< 2- 0 <s<t, seD y (t), mf Y{a)-0 0 <s<t, seD In (3.69) we let N —*• oo to obtain: E In (3.70) we let n —*• oo to get: E Let a > 0 be arbitrary. From (3.71) it follows that < 2" = 0. (3.70) (3.71) Y(t) > a, inf Y (s) = 0 < -E Y(t ), inf Y(s) = 0 0<s<t,seD a 0 <s<t,seD = 0 . Then in (3.72) we let a j 0 to complete the proof of Lemma 3.38. (3.72) □ Let {P(t) : t ^ 0} be a Feller-Dynkin semigroup acting on Cq(E) where FI is a locally compact Hausdorff space. In the proof of Theorem 3.37 we will also use the resolvent operators {R(a) : a > 0}: R(a)f(x) = ^ e~ at P{t)f{x) dt, a > 0, / e Co (FI). An important property is the resolvent equation: R(fi) — R(a) = (a — f3) R(a)R(/3), a, f3 > 0. The latter property is a consequence of the semigroup property. 3.39. Remark. The space E is supposed to be a second countable (i.e. it is a topological space with a countable base for its topology) locally compact Hausdorff space (in particular it is a Polish space). A second-countable locally- compact Hausdorff space is Polish. Let (t7 i ) i be a countable basis of open subsets with compact closures, choose for each i e N, yi e U t , together with a continuous function f, : E —*■ [0,1] such that f, (y 2 ) = 1 and such that f, (y) = 0 for y $ Ui. Since a locally compact Hausdorff space is completely regular this choice is possible. Put OO d(x,y ) = \M X ) ~ Mv)\ + i=l This metric gives the same topology, and it is not too difficult to verify its completeness. For this notice that the sequence (f i ) i separates the points of E, and therefore the algebraic span (he. the linear span of the finite products of the functions /)) is dense in Cq(E ) for the topology of uniform convergence. A proof of the fact that a locally compact space is completely regular can be found in Willard [ 152 ] Theorem 19.3. The connection with Urysohn’s metrization theorem is also explained. A related construction can be found in Garrett [ 57 ]: see Dixmier [ 39 ] Appendix V as well. 3.40. Remark. Next we present the notion of Skorohod space. Let D ([0,1], R) be the space of real-valued functions u> defined on the interval [0,1] that are right-continuous and have left-hand limits, i.e., u>(t) = u> (£+) = lim s j t a;(s) for all 0 < t < 1, and u> (t—) = lim s f t u(s) exists for all 0 < t ^ 1. (In probabilistic literature, such a function is also said to be a cadlag function, “cadlag” being an E=i 2-7i(*) ZZ12 x, y e E. 122 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales acronym for the French “continu a droite, limites a gauche”.) The snpremnm norm on D ([0,1], M), given by IMloo = sup |w(*)|, we5([0,l],I), te[0,l] turns the space D ([0,1],M) into a Banach space which is non-separable. This non-separability causes well-known problems of measurability in the theory of weak convergence of measures on the space. To overcome this inconvenience, A.V. Skorohod introduced a metric (and topology) under which the space be¬ comes a separable metric space. Although the original metric introduced by Skorohod has a drawback in the sense that the metric space obtained is not complete, it turned out (see Kolmogorov [ 70 ]) that it is possible to construct an equivalent metric (he., giving the same topology) under which the space D ([0,1], M) becomes a separable and complete metric space. Such metric space the term Polish space is often used. This metric is defined as follows, and taken from Paulauskas in [ 110 ]. Let A denote the class of strictly increasing continuous mappings of [0,1] onto itself. For A e A, let sup OsSsCtsSl log \(t) - A(s) t — s Then for uq and u) 2 e D ([0,1], E) we define d(ui,u 2 ) = inf max(||A||, Hcui - u; 2 ° tU • The topology generated by this metric is called the Skorohod topology and the complete separable metric space D ([0,1],M) is called the Skorohod space. This space is very important in the theory of stochastic processes. The general theory of weak convergence of probability measures on metric spaces and, in particular, on the space D ([0,1], R) is well developed. This theory was started in the fundamental papers like Chentsov [ 33 ], Kolmogorov [ 70 ], Prohorov [ 112 ], Skorohod [ 122 ]. A well-known reference on these topics is Billingsley [ 17 ]. Generalizations of the Skorohod space are worth mentioning. Instead of real¬ valued functions on [0,1] it is possible to consider functions defined on [0, go) and taking values in a metric space E. The space of cadlag functions obtained in this way is denoted by D ([0, go), E ) and if E is a Polish space, then D ([0, oo ),E), with the appropriate topology, is also a Polish space, see Ethier and Kurtz [ 54 ] and Pollard [ 111 ], where these spaces are treated systematically. Outline of a proof of Theorem 3.37. Firstly, the Riesz representa¬ tion theorem, applied to the functionals / >—► P(t)f(x ), / e Cq(E), (t,x) e [0, oo) x E, provides a family of sub-probability measures B > P (t, x, B). B 6 £, (t, x) e [0, oo) x E, with P(0,x,B) = S X (B) = 1 n(x). From the semi¬ group property, i.e. P(s + t) = P(s)P(t), s, t ^ 0, it follows that the family {P (t, x, ■) : (t, x ) 6 [0, oo)} obeys the Chapman-Kolmogorov identity: P (s + t,x, B) = I P (t, I/, B) P (s, x, dy ), Bef, s>0, t ^ 0, x e E. (3.73) 123 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales The measures B > P(t,x,B), B e £, are inner and outer regular in the sense that, for all Borel subsets B (he. Bef), P (t, x, B ) = sup {Pit, x,K) : K c= B, K compact} = inf {P (t, x, O) : O => B, O open} . (3.74) In general we have 0 < P(t,x,B ) < 1, ( t,x,B ) e [0, go) x E x £. In order to apply Kolmogorov’s extension theorem we need that, for every x e E, the function t >—► P(t, x, E) is constant. Since P( 0, x, E) = 1 this constant must be 1. This can be achieved by adding an absorption point A to E. So instead of E we consider the state space E A = E u {A}, which, topologically speaking, can be considered as the one-point compactification of E , if E is not compact. If E is compact, A is an isolated point of E A . Let £ A be the Borel field of E A . Then the new family of probability measures {N (t, x, •) : (t, x) e [0, oo) x E } is defined as follows: N (t, x, B) = P (t, x, B n E) + (1 — P (t, x, E)) 1 B (A), (t, x) e [0, oo) x E , N (t, A, B) = 1 B (A), t>0, Be£ A . (3.75) www.sylvania.com We do not reinvent the wheel we reinvent light. Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges. An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day. OSRAM SYLVAN!A Light is OSRAM 124 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Compare this construction with the one in (1.1). The family N := {N (t, x, •) : (t, x) e [0, go) x E } again satisfies the Chapman-Kolmogorov identity with state space E A instead of E. Notice that N (t, x, B) = P (t, x, B ) whenever (t, x, B ) belongs to [0, oo )xEx £. Employing the family N we define a family of probability spaces as follows. For every x 0 e E, and every increasing n-tuple 0 < t\ < ■ ■ ■ < t n < go in [0, ) n we consider the probability space ((.E A ) , ® n £ A , P Xo ,ti,...,t n ) : the probability measure P Xo ,t 1 ,...,t n i s defined by P x 0 ,ti,...,t n (-B) = J*. .. fiV(ti -to,x 0 ,dxi) ■ ■ ■ N (t n -t n -i,x n -i,dx n ) , (3.76) B where B e ® n £ A . By an appeal to the Chapman-Kolmogorov identity it follows that, for xo e E fixed, the family of probability spaces: { ((E A ) n , ® n £ A , Px 0 M,..,tn) : o < h < • • • < t n < 00, n e N} (3.77) is a projective system of probability spaces. Put Cl A = (E A ) f 0 ’ 00 ^ and equip this space with the product rr-field U A := ®1°’ 00 )£ A . In addition, write X(t)(u) = uj(t), i9 t cj(s) = cj(s +1), s, t ^ 0, <jJ e Cl A . The variables X(t), t ^ 0, are called the state variables, and the mappings t ^ 0, are called the (time) translation or shift operators. By Kolmogorov’s extension theorem there exists, for every x e E A , a probability measure P„, on the u-field T A such that P* *(<!),■•• ,X(t n ) )eB = P, X,t\ ,...,^77, [B]- (3.78) In (3.78) B belongs to ® n £ A , and 0 < t\ < • • • < t n < oc is an arbitrary increas¬ ing n-tuple in [0, oo). Another appeal to the Chapman-Kolmogorov identity and the monotone class theorem shows that the quadruple {(d a ,T a ,P x ), (x(t),t ^ o) , ($t, t ^ 0), (E A , £ a ) | (3.79) is a Markov process relative to the internal history, i. e. re T A = a fx(s) : 0 ^ s ^ t\ t ^ 0. Moreover, we have P x by the Markov property, we also have, for x e E A , t > s ^ 0, ative to the filtration X(0) = x = 1, and, P, X(t) = A, X(s) = A (Markov property) E, = E, = E, P a | ~X(t - s) otf s A T A ,X(s) = A [X(t-s) = A],X(s) = A P/ X(t A ,X(s) = A = N (t — s, A, {A}) • N (s, x, {A}) N (s, x, {A}) = P a A (3.80) The equality in (3.80) says that once the process t i—> X(t) enters A it stays there. In other words A is an absorption point for the process X. Define, for t > 125 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 0, the mapping P(t) : C (E A ) - C (. E A ) by P(t)f(x) = E x [f (X(f)JJ, / e C ( E A ). From the Markov property of the process X, and since the semigroup 1 > P(t) is a Feller-Dynkin semigroup, it follows that the mappings P(t), t > 0, constitute a Feller (or Feller-Dynkin semigroup) on C ( E A ). Consequently, for any f e C ( E A ), and any to P 0, we have lim sup E x s,t^>t 0 ,s,t^o xeE A f A'(() - / (X( s = 0. (3.81) Let D be the collection of non-negative dyadic rational numbers. Since the space E a is compact-metrizable, it follows from (3.81) that, for all x e E A , the following limits lim X(s), and lim X(s), sft, seD s[t,seD exist in E A P^-almost surely. Define the mapping tt : fl A —> fi luy ir(u)(t) = lim X(s) (a;) =: X(t) (7r(w)), t ^ 0, a;eH A . sit,seD (3.82) (3.83) Then we have that, for every x e E A fixed, the processes t X{t) o it and t X(t) are P x -indistinguishable in the sense that there exists an event D A,/ c 0 A such that P., p A ’' = 1 and such that for all t e [0,c») the equality X(t) = X(t)oTT holds on the event Q Aj . This assertion is a consequence of the following argument. For every (t,x) e [0. x) x E A and for every / e C (E A j we see E* f(X(t)0 7T)-f(X(t lim E t sp, seD f X(s))-f X(t = e t f X(t))-f X(t = 0. (3.84) Since the space E A is second countable, the space C ( E A ) is separable, the equalities in (3.82) and (3.84) imply that, up to an event which is P x -negligible X(t) = X(t) o tt for all t ^ 0. See Definition 5.88 as well. In addition, we have that, for u) e D the realization t h-»- ir(uj)(t) belongs to the Skorohod space D, i.e. it is continuous from the right and possesses left limits. We still need to show 0 < s < t, s 6 d\ {V(s) : that X(t) e E implies that the closure of the orbit is a closed, and so, compact subset of E. For this purpose we choose a strictly positive function /e Co (E) which we extend to a function, again called /, such that / (A) = 0. It is convenient to employ the resolvent operators R(a), a > 0, here. We will prove that, for a > 0 fixed, the process t > e~ at R (a) f (x(t is P x -supermartingale relative to the filtration t\ ^ 0. Then we write: t ^ OF Therefore, let to > E = E. [e-^R(a)f (X («) T, A 0 —at2 f S E X(t 2 ) f(X(s ds Tr 126 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales (Markov property) = E„ f e~ at2 I e f [X (s +i 2 t2 ds I 3 ti (Fubini’s theorem and tower property of condiitonal expectation) E„ = E„ — at 2 J J e as f (x (s + f 2 )) ds | LF; e- as f(X(s ) ds I ^ (the function / is non-negative and £2 > h) < E t = e t fe-/ (X(s)) ds|§) -Jti —at 1 fe~ as f (- X(s + t 1)) ds| (Fubini’s theorem in combinaton with the Markov property) r*oo = e e““ s E Jo —at 1 'X(ti) / (if (s))] ds = e- a, 'R(a)f(x («,)). (3.85) 360 ° thinking Deloitte Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. 127 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Put Y(t) = e at R(a)f (t)^, and fix x e E. From (3.85) we see that the process t > Y(t) is a P x -supermartingale relative to the filtration ) ~ ^ V / t>o From Lemma 3.38 with Y ( t ) instead of Y ( t ) and Ph in place of P we infer X(t) e E, inf Y(s) = 0 seDn(0,t) = Pr Y(t) > 0. inf y( s ) = 0 seDn(0,t) = o. (3.86) From (3.86) we see that, P x -almost surely, X(t) e E implies inf s6l ) n (o > t) Y (s) > 0. Consequently, for every x e E, the equality X(t) 6 E = P x X(t) 6 E, closure jx(s) : s e D n (0, t)\ a E , (3.87) holds. In other words: the closure of the orbit jx(s) : s e D n (0, £)j is con¬ tained in E whenever X (t) belongs to E. We are almost at the end of the proof. We still have to carry over the Markov process in (3.79) to a process of the form m Px) , > 0), (& t , t> 0), (E, £)} (3.88) with fl = D ([0, oo), P A ) the Skorohod space of paths with values in E A . This can be done as follows. Define the state variables X(t) : D —> E A by X(t)(u)) = u)(t), uj e D, and let d t : Q —*■ Q be defined as above, i.e. tf t (uj)(s) = cj(s + 1), u e Cl. Let the mapping n : Cl —> Cl be defined as in (3.83). Then, as shown above, for every x e E, the processes X(t) and X(t) on are ^-indistinguishable. The probability measures P x , x 6 E, are defined by P x [A] = P x \k e A] where A is a Borel subset of Cl. Then all ingredients of (3.88) are defined. It is clear that the quadruple in (3.88) is a Markov process. Since the paths, or realizations, are right-continuous, it represents a strong Markov process. This completes an outline of the proof of Theorem 3.37. □ As above L is a linear operator with domain D(L ) and range R(L) in Cq{E). Suppose that the domain D(L) of L is dense in Cq(E). The problem we want to address is the following. Give necessary and sufficient conditions on the operator L in order that for every x 6 E there exists a unique probability measure P x . on T with the following properties: (i) For every / e D(L) the process f(X(t)) — f(X( 0)) — ^ Lf(X(s))ds, t ^ 0, is a P x -martingale; (ii) P*(X(0) = x) = l. Here we suppose Cl = D ([0, go) , E A ) (Skorohod space) and T is the a-field gen¬ erated by the state variables X(t), t ^ 0. Let P(Cl) be the set of all probability measures on T and define the subset P'(Cl ) of P(Cl) by P'(SJ) = U P 6 P(Cl) : P[X(0) = x] = 1 and for every / e D(L ) the process 128 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales f(X(t)) — f(X( 0)) — f Lf(X(s))ds, t ^ 0, is a P-martingale Jo (3.89) Let (vj : j e N) be a sequence of continuous functions defined on E with the following properties: (i) v 0 = 1; (ii) b-L ^ 1 and Vj belongs to D(L) for j ^ 1; (iii) The linear span of Vj , j ^ 0, is dense in C(E A ). In addition let (/& : k e N) be a sequence in D(L) such that the linear span of {( fk , Lfk) : k e N} is dense in the graph G(L ) := {(/, Lf ) : / e D(L)} of the operator L. Moreover let {s 3 : j e N) be an enumeration of the set Q n [0, oo). The subset P'(O-) may be described as follows: 00 00 00 p, (! 2 ) - n n n n n (3.90) n=lk=lm=l 0 ^ <...<s jjn+1 P e P(fl) : inf max xeE l^j^n J (h(X(s im „)) ~ LMX{s))ds\ n” (V(sj,))dP =j - p LMx^d^j nr,, (vtsjj)®}. It follows that P'(fi) is a weakly closed subset of P(Q). In fact we shall prove that, if for the operator L, the martingale problem is uniquely solvable, then the set P'(fi) is compact metrizable for the metric rf(Pi,P 2 ) given by d(p 1 ,p 2 ) = 2 AcN,|A|<oo 2-W (X( S ,,))d(P 2 (3.91) The following result should be compared to the comments in 6.7.4. of [133]. It is noticed that in Proposition 3.41 below the uniqueness of the solutions to the martingale problem is not used. 3.41. Proposition. The set P'(fl) supplied with the metric d defined in (3.91) is a compact Hausdorff space. Proof. Let (P n : n e N) be any sequence in P'(Q). Let (P n<! : £ e N) be a subsequence with the property that for every me N, for every m-tuple (j i,... ,jm) in and for every m-tuple (,Sj l ,..., Sj m ) e Q m the limit J Cl Vjk ( X dFne 129 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales exists. We shall prove that for every me N, for every m-tuple (j i,..., j m ) in N m and for every m-tuple (tj 1 ,..., tj m ) e [0, oo) m the limit J nr=i Uh ( x dFng ( 3 - 92 ) exists for all sequences (v,j : j e N) in Cq(E). But then there exists, by Kol¬ mogorov’s extension theorem, a probability measure P such that lim r— >oo u '3k {t jk )) d¥ ni (3.93) for all m e N, for all (ji,..., j m ) e and for all (tj 1 ,..., t Jm ) e [0, co) m . From the description (3.89) of P'{ fl) it then readily follows that P is a member of P'(Q). So the existence of the limit in (3.92) remains to be verified together with the fact that D([0,co) , E A ) has full P-measure. Let t be in Q. Since, for every j e N, the process Vj(X(s)) — vj(X(0)) — ^ Lvj(X(a))da, s ^ 0, is a martingale for the measures P n£ , we infer J J Lvj(X(s))dsdP ni = J Vj(X(t))dP ne - J Vj{X{ti))dF nr and hence the limit lim^oo $ Jq Lvj(X (s))dsdF ni exists. SIMPLY CLEVER SKODA We will turn your CV into an opportunity of a lifetime Do you like cars? Would you like to be a part of a successful brand? We will appreciate and reward both your enthusiasm and talent. Send us your CV. You will be surprised where it can take you. 130 Send us your CV on www.employerforlife.com Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Next let t 0 be in [0, go). Again using the martingale property we see V j (*(*>)) d (Png — Pn fc ) - J ([ Lv, (X{ S )) els'] d (P„, - P„J + J v,(X(0))d (P„, - P„J - J ^Lv J (X( a ))ds'jd(P n ,-P, t ), where t is any number in Q n [0, go). From (3.94) we infer Ju i (X(i 0 ))d(P„, -P„J J Q%(X( S ))d«)d(P B< -P B J + j Vj (X(0))d(F ne -F nk ) (3.94) + 2 \t — fo| ||Lvj ||op • If we let i and k tend to infinity, we obtain lim sup >00 jA (X(t 0 )) d (P re£ -P„J ^ 2 \t — t 0 1 I Lv J II 00 ' (3.95) (3.96) Consequently for every s ^ 0 the limit liin^ rx , §Vj (X(,s)) dF ne exists. The inequality il Lvj (X(s)) dsdF ne < l*“*o| \\LVj loo shows that the functions t •—» lirrp^r X , J v,j (X(t)) dP ne , j e N, are continuous. Since the linear span of (vj : j E 1) is dense in Cq(E), it follows that for v e C()(E) and for every t E 0 the limit lim .£—>00 jv(X(t))dF ne (3.97) exists and that this limit, as a function of t, is continuous. The following step consists in proving that for every to e [0, oo) the equality lim lim sup j | vj (X(t)) — Vj (X(t 0 ))| dF ne = 0 (3.98) t_> *o r—>oo J holds. For t > s the following (in-)equalities are valid: (J lv, (X(t)) - V, (X(»))| dP„,) ' J \Vj (X(t)) - V, (X( S ))| 2 dP n , -f M xm’dP v -f M xm^ - 2 Re J (vj(X (()) - v,(X(s)))v,(X(s))dP ni 131 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales — 2Re J Q LvjiXiafidaJvjiXWdPnt < J \vj(X(t))\ 2 dF ne - J \vj(X(s))\ 2 dP n£ + 2 (t - s ) \\L Vj ^ . (3.99) Hence (3.97) together with (3.99) implies (3.93). By (3.93), we may apply Kolmogorov’s theorem to prove that there exists a probability measure P on Cl' := (E A )[° ,co ) with the property that r m In J k =1 v jk( x ( s j k )W = } im r m In J k =1 ■ nt ■> (3.100) holds for all m e N and for all (sj 1 ,..., s 3m ) e [0, co) m . It then also follows that the equality in (3.100) is also valid for all m-tuples /i,..., f m in C(E A ) instead of v ]l ,..., Vj m . This is true because the linear span of the sequence (v 3 : j e N) is dense in C(E A ). In addition we conclude that the processes f(X(t)) — f(X( 0)) — Lf(X(s))ds, t ^ 0, / 6 D(L ) are P-martingales. We still have to show that T>([0, go) , E a ) has P-measure 1. From (3.98) it essentially follows that set of w £ ( E a ) 1° :X ) for which the left and right hand limits exist in E A has ’’full” P-measure. First let / ^ 0 be in Co(E). Then the process [ G\f ] ( t ) : = Eff e- x °f(X{a))da\3 t ) is a P-supermartingale with respect to the hltration {3y : t ^ 0}. It follows that the limits linpp 0 [G\f] ( t ) and lim t p 0 [G\f] ( t ) both exist P-almost surely for all to ^ 0 and for all / e 6' 0 (E). In particular these limits exist P-almost surely for all f e D(L ). By the martingale property it follows that, for fe D(L), \f(X(t))-Xe M [G\f] (t)| = Ae At E( I e- x °(f(X(a))-f(X(t)))da\? t Ae At E (I (f •-“(!' Lf(X(s))ds Ida \ % pOO < AgAt \ t e ~ Xa ( a - Halloo da = A_1 \\ L f\L ■ Consequently, we may conclude that, for all s, t ^ 0, I/(X(t)) - /(V( S ))| « 2A- 1 ||t/L + \\e xt [G\f] (() - \e Xs [G,f] ( s )| and hence that the limits lim^ s f(X(t)) and limq s f(X(t)) exist P-almost surely for all / e D(L). By separability and density of D(L) it follows that the limits lim^ s X(f) and lim ( f s X(t) exist P-almost surely for all s ^ 0. Put Z(s)(uj) = lim 4 Sjt6 QX(t)(c(j), t 5= 0. Then, for P-almost all uj and for all s ^ 0, Z(s)(u>) is well-defined, possesses left limits and is right continuous. In addition we have E(/(ZM)9M) -E(f(X(s+))g(X(s))) -= limE (f(X(t))g(X („))) t[s = E tf{X{s))g{X(s ))), for all f,ge C 0 (E ) and for all s ^ 0: see (3.98). But then we may conclude that X(s) = Z(s) P- almost surely for all s ^ 0. Hence we may replace X with Z and consequently 132 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales (see the arguments in the proof of Theorem 9.4. of Blumenthal and Getoor [[20], p. 49]) P (uj e O ': oj is right continuous and has left limits in £ ,A ) = 1. Fix s > t. We are going to show that the set of paths c o e (.EA)^’ 00 ) for which u>(s) = X(s)(ui) belongs to E and for which, cn(t— ) = lim r f t X(t)(u) - A possesses P-measure 0. It suffices to prove that, for / e Co(E), 1 ^ f(x ) > 0 for all x e E fixed, the following integral equalities hold: E [/(A»), f(X(t-)) = 0] = J/(X( S ))l l/ _ 0 ,(X(t-)) dP = 0. MAERSK I joined MITAS because I wanted real responsibility The Graduate Programme for Engineers and Geoscientists www.discovermitas.com Real work International opportunities Three work placements a I was a construction supervisor in the North Sea advising and helping foremen solve problems 133 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales This can be achieved as follows. From the (in-)equalities E(f(X(s)),f(X(t))=0) = lim E (f(X( S )) (l - (f(X(t))) 1/n )) = limE(/(A'(s)) (1 -/(X(t)) 1 '")) "l/n i = lim E n—> oo rl/n /(A(s))J f(X(t)Y log = lim n—> oo = lim n —»go J " E f/(X(s))f(X(t)r log da Tom) J " (E - E„,) (f(X( S ))f(X(t)r log da 25 , { 7 E "' (f(X(s))f(X(t)T log yplyyy) da £K f(x(s))nx(t)r log + lim n < nm). + E„,(/(A(s)),/(X(f)) = 0), we conclude that E (f(X(s)),f(X(t)) = 0) = lim E (f{X(s)) (l - /(X(f)) 1 /")) j;K/(A W )/ Wt )rio g7 ^) -E ni (nX(s))f{X(t)yi 0t j^rr'\ da. Since the function x >—> f(x) a log belongs to Cq{E) for every a > 0, we / \ x ) obtain upon letting £ tend to oo, that E (f(X(s)), f(X(t)) = 0) = 0, where s > t. To see this apply Scheffe’s theorem (see e.g. Bauer [[ 10 ], Corollary 2.12.5. p. 105]) to the sequence a >-+ E n/; (^f (X (s)) f (X (t)) a log From description (3.90), it then follows that P belongs to P'(Q): it is also clear that the limits in (3.92) exist. □ 3.42. Proposition. Suppose that for every x e E the martingale problem is uniquely solvable. Define the map F : P'(0) —» E A by F( P) = x, where P 6 P'(D) is such that P(X(0) = x) = 1. Also notice that F(Pa) = A. Then F is a homeomorphism from P'ifit) onto E A . In fact it follows that for every u e Co(E) and for every s ^ 0 , the function x >—> Ea;(w(X(s)) belongs to Cq{E). Proof. Since the martingale problem is uniquely solvable for every x e E the map F is a one-to-one map from the compact metric Hausdorff space P'Xl) 134 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales onto E a (see Proposition 3.41). Let for x e E the probability F x be the unique solution to the martingale problem: (i) For every / e D(L) the process f(X(t)) - f(X( 0)) - g Lf(X(s))ds, t ^ 0, is a P x -martingale; (ii) P*(X(0) = x) = 1. Then, by definition F(F X ) = x, x e E, and F(Pa) = A. Moreover, since for every x e E the martingale problem is uniquely solvable we see P'( f2) = {P x : x e E a }. Let (xg : £ e N) be a sequence in E A with the property that lim^oo d (P X£ , P x ) = 0 for some x e E A . Then lim^oo \ v j (%i) ~ v j { x )| = 0, for all j e N, where, as above, the span of the sequence (vj : j e N) is dense in C (E A ). It follows that liiii^x, xg = x in E A . Consequently the mapping F is continuous. Since F is a continuous bijective map from one compact metric Hausdorff space P'(Q) onto another such space E A , its inverse is continuous as well. Among others this implies that, for every s e Q n [0, go) and for every j ^ 1, the function x > §vj (X(s)) dP x belongs to C$(E). Since the linear span of the sequence (vj : j A 1) is dense in Cq(E) it also follows that for every v e Cq(E), the function x i—> (X(s)) dP x belongs to Cq(E). Next let s 0 > 0 be arbitrary. For every j ^ 1 and every s e Q n [0, go), s > sq, we have by the martingale property: sup |E X (vj(X(s))) - E x (^(A(s 0 )))| = sup f E x (Lvj (X(a))) da XEE XEE Jsq ^ (s-So) Halloo . (3.101) Consequently, for every s 6 [0,oo), the function x h-> E x (vj (X(s))), j A 1, belongs to Cq(E). It follows that, for every v e Cq(E) and every s ^ 0, the function x >—► E x (n(X(s))) belongs to Co(E). This proves Proposition 3.42. □ The proof of the following proposition may be copied from Ikeda and Watanabe [61], Theorem 5.1. p. 205. 3.43. Proposition. Suppose that for every x e E A the martingale problem: (i) For every f e D(L) the process f(X(t)) - f(X( 0)) - J ]Lf(X(s))ds, t ^ 0, is a P -martingale; (ii) P(X(0) = x) = l, has a unique solution P = P x . Then the process m Px), (X(t) :t>0),(fl t :t>0),(F,£)} is a strong Markov process. PROOF. Fix xeE and let T be a stopping time and choose a realization of A [1 a ° $t | Tt] , A e 3 r . Fix any oj e O for which A >—> Q y (A) := E x [1^ o d T | $t\ ( w ), 135 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales is defined for all A e £F. Here, by definition, y = Notice that, sice the space E is a topological Hausdorff space that satisfies the second countability axiom, this construction can be performed for P^-almost all u. Let / be in D(L) and fix t 2 > ti ^ 0. Moreover fix C e 3 tl . Then ^ 1 ((7) is a member of 3y 1+ x- Put M,(t ) = f(X(t)) - f(X( 0)) - f 0 Lf(X(s))ds, 0. We have E„ lo) = E„ (M,(h )\ c ). (3.102) We also have J (f(X{t,)) - f(X( 0)) - J) Lf(X(s))ds\ 1 c dQ, f (X(t 2 + T))~ f(X(T)) - Pl/ (X(s + T)) Jo = E. = E. = E rt2+T \ f (X(t 2 + T)) - f(X(T )) - Lf (X(s)) ds) (l c o i9 T ) rt2+T f (X(t 2 + T)) - /(X(T)) - Lf (X(s)) ds u). E, (3.103) ds ) 1(7 o dx | 3^t (w) (w) %1+T 1 c ° | 3"T Because achieving your dreams is your greatest challenge. IE Business School’s Master in Management taught in English, Spanish or bilingually, trains young high performance professionals at the beginning of their career through an innovative and stimulating program that will help them reach their full potential. Choose your area of specialization. Customize your master through the different options offered. Global Immersion Weeks in locations such as London, Silicon Valley or Shanghai. Because you change , we change with you . www.ie.edu/master-management mim.admissions@ie.edu f # In YnTube ii Master in Management • 136 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales By Doob’s optional sampling theorem, the process rt-\-T f {X{t + T)) - f(X(T )) - Lf {X{s)) ds is a P ; ,;-inartingale with respect to the fields %+t, t P 0. So from (3.103) we obtain: J (V(X(t 2 )) - f(X( 0 )) - £ 2 Lf(X(s))ds ) 1 cdQy = E* (j (X(n + T)) - f(X(T )) - £ 1+T Lf (X(s)) , = J (jiXfa)) - f(X( 0 )) - £' Lf(X(s))ds^j 1 cdQy. ds 1 1(7 o dj* | u (3.104) It follows that, for / 6 D(L ), the process Mf(t ) is a P y - as well as a martingale. Since P J/ [X(0) = y\ = 1 and since Qy(*(0) = 2/) = ^ [l{X(0)=2/} ° \ $t\ (X>) = Ex [l{X(T)=y} | 3" T ] (u) = 1 {X(T)=y}(u) = 1, (3.105) we conclude that the probabilities P :y and Q y are the same. Equality (3.105) follows, because, by definition, y = X(T)(u) = uj(T(oj)). Since F y = Q y , it then follows that lPx(T)(a;)(^4) = E x [1 A ° | 3V] (^), A G T. Or putting it differently: Ex(t)(^4) = Eh [1^ o d T \ ff T ], A e T. (3.106) However, this is exactly the strong Markov property and completes the proof of Proposition 3.43. □ The following proposition can be proved in the same manner as Theorem 5.1 Corollary in Ikeda and Watanabe [61], p. 206. 3.44. Proposition. If an operator L generates a Feller semigroup, then the martingale problem is uniquely solvable for L. Proof. Let {P(t) : t P 0} be the Feller semigroup generated by L and let m P*), (X(t) :t>0),(# t :t>0),(E,e.)} be the associated strong Markov process (see Theorem 3.37). If / belongs to D(L), then the process M f (t) := f(X(t)) - f(X( 0)) - J \ Lf(X(s))ds , t> 0, is a P ;) ,-martingale for all x e E. This can be seen as follows. Fix f 2 > T P 0. Then E x [M f (t 2 ) | T tl ] = Mfiti) + Ea; j = Mfitf) + Ea; f(x(h)) -1 Lf(x(s))dfj I s-,,) - S(X(U)) f(X(h - h + «) - F " Lf(X(s + U))ds 137 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales - f(X(t i)) (Markov property) Mfih) + E x(tl) - U)) - £ 2 h Lf(X(s))ds^j Next we compute, for y e E and s > 0, the quantity: f(X(t 0). (3.107) E, (j(X(s)) - f Lf(X(cr))d(j S J - f(y ) = [P(s)f] (y) - f [P(a)(Lf)] ( y)da - f(y) Jo = [i=(s)/] (y) - £ Ya [p{(,)s] {v)ch - Hv) = [i=(s)/]] (y) - (Vw/1 (v) - [^(0)/] (!/)) - m = 0. (3.108) Hence from (3.107) and (3.108) it follows that the process t ^ 0, is a P^-martingale. Next we shall prove the uniqueness of the solutions of the martingale problem associated to the operator L. Let P). and P'). be solutions ’’starting” in x e E. We have to show that these probabilities coincide. Let / belong to D(L) and let T be a stopping time. Then, via partial integration, we infer A J” e“ At | f(X(t + T)) - £ +T Lf(X(r))dr - f(X(T)) j dt + f(X(T)) A J°° e" At | f(X(t + T)) - £ +T Lf(X(r))dT | dt rGO rGO rt A e~ M f(X(t + T))dt - A e“ At Lf(X(r + T))drdt Jo Jo Jo rGO rGO / r*< = a| e~ xt f(X(t + T))dt- A I (J rGO = e~ xt [(A/ — L)f] (X(t + T))dt. Jo e~ xt dt ) Lf(X(r + T))dr (3.109) From Doob’s optional sampling theorem together with (3.109) we obtain: rGO <=“X ((A/ - L)f(X(t + T)) I 3V) dt - f(X(T)) Jo = A f° e~ x V, | ff(X(t + T)) - J Lf (X(r))d T - f(X(T))) \J T \dt = 0 = A f° e- A, E 1 1 (f(X(t + T)) - £ Lf(X( T ))dT - f(X(T))\ \J T \dt rGO = e~ xt El ((A/ — L)f{X(t + T)) \ 3?) dt — f(X(T)). (3.110) Jo 138 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Next we set rco [R(\)f] (x) = e~ Xt [P(t)f] ( x)dt , xe E, A > 0, f e C 0 (E). Jo Then (3.111) (XI - L)R(X)f = /, / s C„(E), R(X)(XI -L)f = f,f e D(L). (3.112) Among other things we see that R(XI — L) = Co(E), A > 0. From (3.110) it then follows that, for g e Co(E), r oo r oo e- x X(g(X(t + T))\ J T )dt=\ e~ Xt [P{t)g](X(T))dt Jo Jo f e~ xt E % ( g(X(t + T)) \ J T ) dt. (3.113) Since Laplace transforms are unique, since g belongs to C'o(E) and since paths are right continuous, we conclude Ei (g(X(t + T)) | S T ) - [P(i) s ](X(T)) - E* (g(X(t + T)) | ? T ), (3.114) whenever g belongs to Cq(E), whenever t ^ 0 and whenever T is a stopping time. The first equality in (3.114) holds P^-almost surely and the second P^- almost surely. As in Theorem 3.36 it then follows that k (rC-i m x m~>) = e " (nj-i m x m>) (3.H5) for n = 1 , 2 ,... and for / 1? ..., f n in Cq(E). But then the probabilities P* and P^ are the same. This proves Proposition 3.44. □ 139 Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales The theorem we want to prove reads as follows. 3.45. Theorem. Let L be a linear operator with domain D(L ) and range R(L) in Cq(E). Let Cl be the path space Cl = D ([0, go) , £’ A ). The following assertions are equivalent: (i) The operator L is closable and its closure generates a Feller semigroup; (ii) The operator L solves the martingale problem maximally and its domain D(L) is dense in Co(E); (iii) The operator L verifies the maximum principle, its domain D{L ) is dense in Cq(E) and there exists Ao > 0 such that the range R (Ao I — L ) is dense in Cq{E). 3.46. Remark. The hard part in (iii) is usually the range property: there exists A 0 > 0 such that the range i?(A 0 / — L ) is dense in Cq(E). The theorem also shows, in conjunction with the results on Feller semigroups and Markov pro¬ cesses, the relations which exist between the unique solvability of the martingale problem, the strong Markov property and densely defined operators verifying the maximum principle together with the range property. However if L is in fact a second order differential operator, then we want to read of the range property from the coefficients. There do exist results in this direction. The interested reader is referred to the literature: Stroock and Varadhan [133] and also Ikeda and Watanabe [61]. In what follows we shall assume that the equivalence of (i) and (iii) already has been established. A proof can be found in [141], Theorem 2.2., p.14. In the proof of (ii) => (i) we shall use this result. We shall also show the implication (i) => (ii)- PROOF, (ii) => (i). Let, for x 6 E, the probability P,„ be the unique solution of the martingale problem associated to the operator L. From Proposition 3.43 it follows that the process {(12, T, P x ), (X(t) : t ^ 0), (i9 t : t ^ 0), (E , £)} is a strong Markov process. Define the operators {P(t) : t ^ 0} as follows: [P(t)f](x)=E x (f(X(t))), feCo(E), t> 0 . (3.116) We also define the operators {R( A) : A > 0} as follows: rCO [i?(A)/](x) = e~ M [P(t)f](x)dt, feC 0 (E), A > 0. (3.117) do From Proposition 3.42 it follows that the operators P(t) leave C${E ) invariant and hence we also have R(X)Cq(E) cr C () {E). From the Markov property it fol¬ lows that {P(t) : t A 0} is a Feller semigroup and that the family {R( A) : A > 0} is a resolvent family in the sense that P(s + t) = P(s) o P(t), s , t ^ 0, (3.118) R(A 2 ) — i?(Ai) = (Ai — A 2 ) -R(Ai) o i?(A 2 ), Ai, A 2 > 0. (3.119) For A > 0 fixed the operator L is defined in G'o( E) as follows: L:R(\)f~\R(\)f-f, feC 0 (E). (3.120) 140 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Here the domain D(L ) is given by D(L ) = {R(X)f : f e Cq(E)}. The operator L is well-defined. For, if fi and / 2 in G,( E) are such that R(X)f\ = R(X)f 2 , then by the resolvent property (3.119) we see /ri?(/i)/i = /ii?(/i)/ 2 , /r > 0. Let /i tend oo, to obtain /i = / 2 . Since the operator R(X) is continuous, the operator L is closed. Next we shall prove that L is an extension of L. By partial integration, it follows that, for / e D(L), e“ At | f(X(t)) - f(X( 0)) - £ Lf(X(r))dr + A £ e~ Xs |/(X(s)) - /(*«,)) - £ L/(X(r))dr| ds = e~ Xt f(X(t )) - f(X( 0)) + f e~ Xs (XI - L)f(X(s))ds. (3.121) Jo As a consequence upon applying (3.121) once more, the processes e~ Xt f(X(t )) - f(X( 0)) + £ e~ Xs (XI - L)f(X(s))ds : t> o| , fe D(L), (3.122) are P^-martingales for all x e E. Here we employ the fact that the processes j/(X(f)) - f(X( 0)) - £ Lf(X(s))ds :t> o| , fe D{L ), are P x -martingales. This is part of assertion (ii). From assertion (3.122) it follows that 0 = E x (e~ Xt f{X(t)) - f(X( 0)) + £ e~ Xs (XI - L)f(X(s))ds^j , / g D(L). (3.123) Let t tend to infinity in (3.123) to obtain rco 0 = -E x (f(X(0))) + e~ Xs E x ((XI-L)f(X(s)))ds, f e D(L). (3.124) Jo From (3.124) we obtain f{x) = ^ e~ Xs [P(s)(XI — L)f] (x)ds, f e D(L). Or writing this differently / = R(X)(XI — L)f , / e D(L). Let / belong to D{L). Then / = R(X)g, with g = (A I — L)f and hence / belongs to D(L). Moreover we see If = L(R(X)g) = XR(X)g - g = Xf - (A/ - Lf ) = Lf. (3.125) It follows that L is a closed linear extension of L. In addition we have R(XI — L ) = 6'o (E). We shall show that the operator L verifies the maximum principle. This can be achieved as follows. Let / in Cq(E) be such that, for some x 0 e E, Re (R(X)f)(x 0 ) = sup {Re R(X)f(x) : x e E) > 0. (3.126) Then Re (i?(A)/)(xo) A Re R(X)f(X(t)), 0, and hence rCO Re (R(X)f)(x 0 ) > Re e~ Xs E x{t) (f(X(s))) ds, t> 0. (3.127) Jo 141 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales So that, upon employing the Markov property, we obtain for t ^ 0: rco Re (R(\)f)(xo) » Re I e - A *E„ (E Y(i) (f(X(s)))) ds - Re E„ ([R(A)/] (X(t)). Hence, for /u > 0, we obtain 1 f 00 —Re ( R(X)f)(xo ) = e ^Re R(X)f(xo)dt f 1 Jo ^ Re r oo e-<“E „([R(A)/](A-(t)))* Jo (resolvent equation (3.119) = Re R(n)R(\)f(x 0 ) R(X)f(x 0 ) - R(fj)f(x 0 ) = Re H — X (3.128) (3.129) Consequently: [ARe R(X)f] (x 0 ) < Re [fiR(fi)f] (rr 0 ), /i > A. Let ft, tend to infinity, use right continuity of paths and the continuity of / to infer Re LR(X)f(x 0 ) = Re {XR(X)f(x 0 ) - f(x 0 )} < 0. (3.130) AACSB ACCREDITED Excellent Economics and Business programmes university of groningen www.rug.nl/feb/education “The perfect start of a successful, international career.” CLICK HERE to discover why both socially and academically the University of Groningen is one of the best places for a student to be 142 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales This proves that L verifies the maximum principle. Employing the implication (iii) => (i) in Theorem 3.45 yields that L is the generator of a Feller semigroup. From Proposition 3.44 it then follows that for L the martingale problem is uniquely solvable. Since L solves the martingale problem maximally and since L extends L, it follows that L = L, the closure of L. Consequently the operator L is closable and its closure generates a Feller semigroup. (i) => (ii) Let the closure of L, L, be the generator of a Feller semigroup. From Proposition 3.44 it follows that for L the martingale problem is uniquely solvable. Hence this is true for L. We still have to prove that L is maximal with respect to this property. Let L\ be any closed linear extension of L for which the martingale problem is uniquely solvable. Define L\ in the same fashion as in the proof of the implication (ii) => (i), with Li replacing L. Then L\ is a closed linear operator, which extends L\. So that L\ extends L. As in the proof of the implication (ii) => (i) it also follows that Li generates a Feller semigroup. Since, by (i), the closure of L, also generates a Feller semigroup, we conclude by uniqueness of generators, that L = L\. Since L\ => L\ = L\ => L =2 L, it follows that the closure of L coincides with L\. This proves the maximality property of L, and so the proof of Theorem 3.45 is complete. □ In fact a careful analysis of the proof of Theorem 3.45 shows the following result. 3.47. Proposition. Let L be a densely defined operator for which the martin¬ gale problem is uniquely solvable, and which is maximal for this property. Then there exists a unique closed linear extension L 0 of L, which is the generator of a Feller semigroup. PROOF. Existence. Let {P x : x e E) be the solution for L, and assume that for all / 6 Cq(E) the function x >—> [ P(t)f] (x) belongs to Co(E) for all t ^ 0. Here P(t)f(x) is defined by [/>(()/] (*) - E* (/(V(t))), [B(A)/](! L„(R(\)f) := \R(X)f - f. J r Co(E). Here t 5= 0 and A > 0 is fixed. Then, as follows from the proof of Theorem 3.45, the operator L 0 generates a Feller semigroup. Uniqueness. Let L\ and L 2 be closed linear extensions of L, which both generate Feller semigroups. Let {(D, T, Pi), (X(t) 0), (E, £)} respectively {(D, T, P^, (X(t) :t>O),(0 t :t> 0), (E, £)} be the corresponding Markov processes. For every / e D(L), the process f(X(t)) — f(X( 0)) — Sf 0 Lf(X(s))ds, t ^ 0, is a martingale with respect to as well as with respect to P^. Uniqueness implies Pj, = P| and hence L\ = L 2 . The proof of Proposition 3.47 is complete now. □ 143 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.48. Corollary. Let L be a densely defined linear operator with domain D(L ) and range R(L) in Cq{E). The following assertions are equivalent: (i) Some extension of L generates a Feller semigroup. (ii) For some extension of L the martingale problem is uniquely solvable for every x e E. PROOF, (i) => (ii). Let L 0 be an extension of L that generates a Feller semi¬ group. Let {(f1, ft, P x ) ,{X{t) : t 5s 0), (d t : t ^ 0), ( E , £)} be the corresponding Markov process. For x e E the probability P x is the unique solution for the martingale problem starting in x. (ii) => (i). Let L 0 be an extension of L for which the martingale problem is uniquely solvable for every x e E. Also suppose that L 0 is maximal for this property. Let {P x : x e E] be the unique solution of the corresponding martin¬ gale problem. Define the operators P{t), t ^ 0, by [P(t)f] (x) = E x (f(X(t))), f e Cq(E). From the proof of Theorem 3.45 it follows that {P(t) : t ^ 0} is a Feller semigroup with generator L 0 • This completes the proof of Corollary 3.48. □ 3.49. Example. Let Lq be an unbounded generator of a Feller semigroup in Co(E ) and let ///, and zy : , 1 C k C n, be finite (signed) Borel measures on E. Define the operator L^p as follows: n ( -n k =1 ^ Ljipf = Lof, f e D . Then the martingale problem is uniquely solvable for Lpp. In fact let {(D, T, P x ), (X(t) :t>O),{0 t :t>O),(E,8')} be the strong Markov process associated to the Feller semigroup generated by L 0 . Then P = P x solves the martingale problem D (L f f) (a) For every / - D{Lpp) the process /(*(«)) - f(X( 0)) - f Lpj,f{X(s))ds, t e 0, Jo is a P-martingale; (b) P(X(0) =x) = l, uniquely. This can be seen as follows. We may and do suppose that the func¬ tionals / I—> J Lofdfik ~ $ fdvk, f e -D(L 0 ), 1 ^ k < n, are linearly independent. If some Hk belongs to D (Lf ), then D (Lpp) is not dense and if none of the measures ///,. belongs to D (L(j). then D (Lpv) is dense in Cq(E'). In either case there exists a unique extension, in fact L 0 , of Lpjy which generates a Feller semi¬ group. Therefore we choose functions Uk e D(L 0 ), 1 C k E n. in such a way that J LoUkdne — $ Ukdvt = 5k/, 1 ^ k, £ ^ n. Suppose that P x and P x are prob¬ abilities, that start in x, with the property that for all / e D (Lp,y) the process 144 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales t i-> f{X(t )) — f{X{ 0)) — Sf 0 Lof(X(s))ds is a P*- as well as a P^-martingale. As in (3.110) we see that for all / e D(Lq) (vk = (A/ — L 0 ) Uk, 1 ^ k ^ n): fdv k u k (x) (3.131) f( x ) ~ Yj (J L ° f d ^ k ~ J = £° e^E* I - L 0 ) - f] ( J L 0 fdfi k - J fdvk'j u k j (X(s)) j ds = e“ As E 2 ^(A I - L 0 ) (f - ( J L ofdfik ~ J fdu^ (X(s)) j ds. Write / = (XI — L 0 ) _1 g = R(X)g. From (3.131) we obtain (3.132) R(X)g(x) - 2 (J (XR(\)g ~ 9 ) dn k - J R(X)gdu^j u k (x) = I e -As Ei I f e“ As E? 9 (X(s)) - p (J (XR(X)g - g) d 4 i k - J R(X)gdv^j v k (X(s)) g( x ( s )) ~ p (J (XR(X)g - g ) dg k - J R(X)gdu k ^ v k (X(s)) ds ds. Put F( A) = (Fi (A),..., F n ( A)) and put U(X) = (u k ^( A)), where, for 1 < k < n, rCO F k ( A) = e~ Xs (El [(XI - L 0 )u k (X(s ))] - E 2 [(XI - L 0 )u k (X(s))]) ds, Jo and where u k4 , 1 < k, I < n, is given by « m (A) = Ja RWu^-ju^-jRWu^. Since (3.132) is valid for all g e Cq(E), it follows that F( A) = U(X)F(X). Since, in addition lim^oo^-^) = 0, we see F k (X) = 0 for all A > 0 and for 1 < k < n. So that e _As E^ (u k (X(s))) ds = J* e _As E^ [tifc(X(s))] ds for all A > 0 and for all 1 k n. Again an application of (3.132) yields Ej. [g(X(s))] = E 2 [g(A(s))] for all g e Cq(E). Since these arguments are valid for any x e E, we conclude just as in Proposition 2.9 and its Corollary on page 206 of Ikeda and Watanabe [ 61 ]), that P* = P 2 = F x , x e E, In particular we may take E = [0,1], L 0 f = \f", D(L 0 ) = {/ e C 2 [0,1] : /'(0) = /'(1) = 0}, g, k (I) = 1 i(s)ds, v k = 0, 0 ^ a k < f3 k ^ 1, 1 ^ k ^ n. Then L 0 generates the Feller semigroup of reflected Brownian motion: see Liggett [86], Example 5.8., p. 45. For the operator the martingale problem is uniquely (but not maximally uniquely) solvable. However it does not generate a Feller semigroup. The previous arguments do not seem to be entirely correct. It ought to be replaced with some results in Section 10 (e.g. Theorem 3.110). Problem. We want to close this section with the following question. Suppose that the operator L possesses a unique extension L 0 , that generates a Feller semigroup. Is it true that for L the martingale problem is uniquely solvable? 145 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In general the answer is no, but if we require that L solves the martingale problem maximally, then the answer is yes, provided as sample space we take the Skorohod space. This result is proved in Theorem 3.45. For the time being we will not pursue the Markov property. However, we will continue with Brownian motion and stochastic integrals. First we give the definition of some interesting processes. American online LIGS University is currently enrolling in the Interactive Online BBA, MBA, MSc, DBA and PhD programs: ► enroll by September 30th, 2014 and ► save up to 16% on the tuition! ► pay in 10 installments/2 years ► Interactive Online education ► visit www.ligsuniversity.com to find out more! Note: LIGS University is not accredited by nationally recognized accrediting agency by the US Secretary of Education. More info here. 146 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 4. Martingales, submartingales, supermartingales and semimartingales Let (Q, T, P) be a probability space and let {T t : t ^ 0} be an increasing family of ex-fields in 'J. If necessary we suppose that the filtration {T t : t > 0} is right continuous, i.e. r J t = or complete in the sense that, for every t > 0, the (j-field “J t contains all A e IF, with P(A) = 0. Let {H(t) : t ^ 0} be a collection of E-valued functions defined on 0. Such a family is called a (real-valued) process. 3.50. Definition. The following processes and cr-fields will play a role in the sequel. (a) The process {H(t) : t ^ 0} is said to be adapted or non-anticipating if, for every t ^ 0, the variable H(t) is measurable with respect to A t . (bl The symbol A denotes the cr-field (= a-algebra) of subsets of [0, go) x Q. which is generated by the adapted processes which are right-continuous and which possess left limits. These are the so-called cadlag processes. (b2) The process {H(t) : t ^ 0} is said to be optional if the function (t, uj) —*• H(t,u>) is measurable with respect to A. (cl) The symbol II denotes the a-field of subsets of [0, oo) x Q, which is generated by the adapted processes which are left-continuous adapted processes. (c2) The process {H(t) : t ^ 0} is said to be predictable if the function (t,cu) —> H(t, uj) is measurable with respect to II. 3.51. Proposition. The collection {(a, b] x A: 0 ^ a < b, A e T a } generates the a-field II. Proof. Let A belong to T a . The variable uj « l( a , 6 ](>s) 1.4 (go) is measurable with respect to and the function s >—>■ l(a,fo] (s) l^i(co) is left continuous. This proves that (a, b] x A belongs to II. Conversely let F be adapted and left continuous. Put Then, by left continuity, lim n ^oo F n (s, uj) = F(s,u>), E-almost surely. Moreover the processes (F n (t) : t p 0} are adapted and are measurable with respect to the (j-field generated by {(a, b] xA:0^a<b,Ae T a }. All this completes the proof of Proposition 3.51. □ 3.52. Remark. Since L a q(s) x 1 Auj) = lim lr a ^ )(s)l AaA where a n [ a and where b n [ 6, it follows that II <= A. Here we employ Proposition 3.51. 3.53. Definition. Let {X(t) : t ^ 0} be an adapted process. (a) The family {X(t) : t 5= 0} is a martingale if E (|X(f)|) < go, t ^ 0, and if, for every t > s ^ 0, E(X(t) | T s ) = X(s), P-almost surely. 147 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales (b) The family {X(t) : t ^ 0} is a submartingale if E(|X(t)|) < oo, t ^ 0, and if, for every t > s ^ 0, E (X(t) | £F S ) ^ X(s), P-almost surely. (c) The family {X(t) : t ^ 0} is a supermartingale if E (|X(t)|) < go, t 5= 0, and if, for every t > s 5= 0, E (X(t) | T s ) < X(s), P-almost surely. (d) It is P-almost surely of finite variation (on [0, t]) if sup ^ | X(tj) — X(tj- 1 )| : 0 < to < ti < ... < t n < co j < go, ^sup „ \X(tj) - r 0 « «„ < h s: t} < co,) P-almost surely. (e) It is a local martingale if there exists an increasing sequence of stopping times ( T n : n e N) for which lim n ^oo T n = oo, P-almost surely, and for which the processes {X(T n a t) : t ^ 0}, n = 1,2,... are martingales with respect to the filtration {Tr uA t : t A 0}. (f) Let T be a stopping time. The process {X(t) : t ^ 0} is a local martin¬ gale on [0, T ) if there exists a sequence of stopping times ( T n : n e N) which is increasing for which lim„_> x T n = T, P-almost surely, and for which the processes { X(T n a t) : t ^ 0}, n = 1, 2,... are martingales with respect to the filtration {Tr n A t-t> 0}. (g) The definition of “local submartingale”, “local supermartingale” and “being locally P-almost surely of finite variation” are now self-explan¬ atory. (h) The process {A" (t) : t A 0} is called a semi-martingale if it can be writ¬ ten in the form X(t) = M(t) + A(t), where { M(t ) : t ^ 0} is a mar¬ tingale and where {A(t) : t ^ 0} is an adapted process which is finite variation, P-almost surely, on [0, t] for every t > 0, and for which E\A(t)\ < oo, t ^ 0. (i) The process {X(t) : t ^ 0} is of class (DL) if for every t > 0 the family { X(t) : 0 ^ r ^ t , r is a (T t ) -stopping time} is uniformly integrable. 3.54. Remark. Let {X(t) : t ^ 0} be a semi-martingale. The decomposition X(t) = M(t ) + A{t ), where (M(f) : t ^ 0} is a martingale and where for every t > 0 the process {.4(f) : t A 0} is P-almost surely of finite variation and where {A(t) : f A 0} is predictable and right continuous P-almost surely is unique, provided .4(0) = 0, P-almost surely. This follows from the fact that a right- continuous martingale which is predictable and of finite variation is necessarily constant: this is a consequence of the uniqueness part of the Doob-Meyer de¬ composition: see Theorem 1.24. A proof of the Doob-Meyer decomposition theorem may start as follows. Put and (3.133) Mt)-M «)+ 2 e x (Kfc<2 n k + 1 2 > 148 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales and prove Mj(t) := Xj(t ) — Aj(t) is a martingale. Then let j —> go to obtain: X(t) = M(t) + A(t), where M(t) = lirn^oo Mj (t) and A(t) = lim^oo Aj(t). 3.55. Remark. An Sy-martingale {M(t) : t ^ 0} is of class (DL), an increasing adapted process {A(t) : t ^ 0} in T, P) is of class (DL) and hence the sum { M(t ) + A(t) : t ^ 0} is of class (DL). If {X(t) : t P 0} is a submartingale and if /r is a real number, then the process {max (X(t), p) : t ^ 0} is a submartingale of class (DL). Pro¬ cesses of class (DL) are important in the Doob-Meyer decomposition theorem. We continue with some examples of martingales, submartingales and the like. 3.56. Example. Let T : fl —> [0, co] be a stopping time. Since T is a stopping time and since the process {l{r<q : t 5= 0} is left continuous, it is predictable. It follows that the process (l{i>t} : t ^ 0} is predictable as well. 3.57. Example. Let / be an open interval in M and let </?:/—> (—oo, co) be an increasing convex function. If {X(t) : t ^ 0} is a submartingale with values in /, then the process {</?(X(f)) : t 5= 0} is also a submartingale. For let t > s ^ 0. Then by the Jensen inequality and the monotonicity of cp it follows that E [p(X(t)) \? s ]>p [E (X(t) | T s )] ^ p (X(s )). 3.58. Example. Let (B(t), P 0 ) be one-dimensional Brownian motion starting in 0. Then {B(t) : t ^ 0} is a martingale. Since the definition of martingale also makes sense for vector valued processes, we also see that an valued Brownian motion is a martingale. 3.59. Example. Let (B(t), P 0 ) be E"-valued Brownian motion starting in 0. The process {\B(t)\ 2 — ut : t ^ 0} is a martingale. 3.60. Example. Let {X(t),Pa;} be a (strong) Markov process such that E x [f(X(t))] = J p(t,x,y)f(y)dm(y), f > 0, where the density p(t, x. y ) verifies the Chapman-Kolmogorov identity: p(s + t, x, y) = J p(s, x, z)p(t, z, y)dm(z). The process {pit — s, X(s), y) : 0 < s < t} is a martingale on [0, t). For example for X{t) we may take Bit), d-dimensional Brownian motion. Then p{t, x, y) = p d (t, x, y) = - . exp (V2 Vt) 3.61. Example. Let {X(t) : t ^ 0} be a right-continuous martingale and let T be a stopping time. The process {X (T a t) : t ^ 0} is a martingale with respect to {T t : t ^ 0} and also with respect to the filtration {Ttaj : t ^ 0}. 149 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.62. Example. This is a standard example of a closed martingale, i.e. a martingale which is written as conditional expectations on cr-fields taken from filtration. Let Y be an random variable in L 1 P). The process s i—► [Y | 3s], s ^ 0, is a martingale. We want to insert an inequality on the second moment of a martingale. This is a special case of the Burkholder-Davis-Gundy inequality. 3.63. Proposition. Let { M(t ) : t ^ 0} be a continuous martingale with M( 0) = 0. Then E (M(t) 2 ) < E (M*(t) 2 ) ^ 4E (M(t) 2 ) . Here M*(t ) = sup 0 ^ t |M(s)|. A cate-Lucent www.alcatel-lucent.com/careers What if you could build your future and create the future? One generation’s transformation is the next’s status quo. In the near future, people may soon think it’s strange that devices ever had to be “plugged in.” To obtain that status, there needs to be “The Shift”. 150 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Proof. Define for £ > 0 the stopping time by T s = inf{t > 0 : M*(t) > . Then {M*(t) > £} Q {T% < t} and {T \ < t} cr { M*(t ) h £} and hence, since \M(t)\ is a submartingale we obtain upon using Doob’s optional sampling rCO E (M*(t) 2 ) = \ P (M*(t) 2 > A) dX Jo (make the substitution A = £ 2 ) r*00 rCO = 2 >£)<%< 2 £P(T s <t)d£ Jo Jo rOO = 2 E(|M(T { )|:T { <t)de Jo (Doob’s optional sampling) r*00 <2 E(|M(t)| : T ( < t)d£ Jo rOO = 2 E(\M(t)\:M*(t)>£)d£ Jo = 2E (\M(t)\ M*(t)) (Cauchy-Schwarz’ inequality) < 2 (E (M(t) 2 )) 172 (E (M*(t) 2 )) 1/2 . Consequently E (M*(t) 2 ) ^ 4E(M(t) 2 ). This completes the proof of Proposition 3.63. □ 3.64. Remark. The method of works very well if E(M*(t) 2 ) is finite. If this is not the case we may use a localization technique. The reader should provide the details. Perhaps truncating is also possible. 5. Regularity properties of stochastic processes In Theorem 3.18 we proved that Brownian motion possesses a continuous ver¬ sion. We want to amplify this result. In fact we shall prove that Brownian motion has Holder continuous paths of any order a < |. This means that for every a < \ and for every a > 0, a e R, there exists a random variable C(b), depending on Brownian motion such that for all 0 ^ s < t < o, the inequality | b(t) — 6(s)| ^ C(b ) 1 1 — s|“ holds P-almost surely. This will be the content of Theorem 3.67 below. We begin with a rather general result, due to Kolmogorov, for arbitrary stochastic processes. 151 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.65. Theorem. Fix a finite interval [a, b]. Let (X(s) : a < s < b] be a stochas¬ tic process on the probability space (f2,£F, P). Suppose that there exist constants K, r and p, such that 0 < r < p < oo and such that E (| X(t) - X(s)| p ) ^ K\t - s| 1+r (3.134) for all a ^ s,t < b. Fix 0 < a < r/p. Then there exists a random variable C(X), which is finite P -almost surely, such that \X(t)-X(s)\^C(X)\t-s\ a (3.135) for all dyadic rational numbers s and t in the interval [a, b]. In particular it follows that a process X = (X(s) : a < s < 6 } verifying (3.134) has a Holder continuous version of order a, a < r/p. PROOF. It suffices to prove (3.135), because the version problem can be taken care of as in Theorem 3.18. Without loss of generality we may and do suppose that a = 0 and that 6 = 1 . Otherwise we consider the process Y defined by T(s) = X ((a 0 + s( 6 0 — a 0 )), 0 < s < 1, where a 0 and 6 0 are dyadic rational with a 0 < a and with b < 6 0 and where outside of the interval [a, 6 ] the process X is defined by X(t) = X(a), if ao < t ^ a, and X(t) = X(b), if 6 < t < bo. Put e = r — ap. Then P(|X(f) -X(s)| > \t-s\ a ) < |f-sp p E(|X(A) -X(s)| p ) < K\t-s\ 1+£ , (3.136) so that Hence X A: + 1 -xX 2 n 00 2 n —l 2 2 P n= 1 k =0 x k+l > 2~ xX -no-ne < K2~ n 2 > 2 " K ^ K V 2 n 2~ n 2~ ne = - *-i 2 e — 1 n=l By the Borel-Cantelli lemma it follows that (3.137) (3.138) un \m—l n'Xm max (K/c^2 n -l x k + 1\ ( k — A < 2~ na U = 1 . (3.139) Hence there exists a random integer u(X) with the following property: For P-almost all u> the inequality max X k + l ^ 2~ na (3.140) is valid for n ^ f(X). Next let n 5= f(X) and let A be a dyadic rational in the interval [A;2 _n , (k + l)2 _n ]. Write t = k2~ n + Xqli eac h 7 j equals 0 or 1. Then X(t) - X k < m Ti 2 <x(n+j) 3 = 1 < 12 " (3.141) 152 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Similarly we have, with t = £2 N , N ^ n, (k + 1)2 n = £2 N + Xljli 7 N ^ 7 ' equals 0 or 1 , X(t) - X k + 1 m 7 j= 1 <: 1 2 ^° <: 1 1 ! - 1 2 no (3.142) Next let s and t be dyadic rationale numbers with 0 < t — s < 2 Take neN with 2 _n_1 ^ t — s < 2~ n and pick k in such a way that k2~ n ~ 1 < s < (fc + l)2 _n_1 . Then (fc + l)2 “ n - 1 ^t = t-s + s < 2~ n + (k + l)2~ n ~ l = (fc + 3)2“ n-1 . It follows that, since t belongs to [(k + l)2 _n_1 , (k + 2)2 _n_1 ] or to the interval [(k + 2)2~ n ~ 1 , (k + 3 ) 2 _n_1 ], |X(t)-X( S )| X(t) - X k + 2 2 n+1 1 _2~{n+l)a < - + 3 I* k + 2 2 n+1 X k + 1 2 n+1 + X k+1 2 n+1 ■ X(s) (3.143) If 1 ^ t — s > 2~^ x \ we choose k and £ e N in such a way that 2+^ > £ > k ^ 0 and that £2~ v<yX ^ < t < (£ + l)2~ u( - x ^ and k2~ v ^ x " > < s ^ (k + l)2~ u( ' X \ Then we get \X(t)-X(s) |< + X(t) - X k £ + 1 2"( x ) + T j=k X 3 +1 2 U ( X ) -X 2 "P 0 X <: 2 L/ ( X ) 2 + 2 " (x) - V(s) 9 _l 2~ai/(X) ^ £±£ - |* _ s |a 2 a — 1 2 a — 1 From (3.143) and (3.144) the result in Theorem 3.65 follows. (3.144) □ In order to apply the previous result to Brownian motion, we insert a general equality for a Gaussian variable X. 3.66. Proposition. Let X : fl —> E be a non-constant Gaussian variable. Then its distribution is given by F(XeB) 1 [ [ 1 k-E(X)| 2 (2ttE(X 2 -(E(X)) 2 )) 172 Jtf 6XP V 2E(X 2 ) - (E(X)) 2 and its moments E (|X — E(X)| P ), p > — l, are given by ^ dx) (3.145) E(|X-E(X)| P ) 2 l^r (Ip + i) 7T V E ( A ' 2 - ( E (V)) 2 ) (3.146) 153 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Proof. Equality (3.145) follows from formula (3.8) and formula (3.146) is proved by using (3.145). The formal arguments read (we write Y = X — E(X)): The latter is the same as (3.146). □ 3.67. Theorem. Let { b(s ) : s > 0} be d-dimensional Brownian motion. This process is P -almost surely Holder continuous of order a for any a < In the past four years we have drilled * 81,000 km A That's more than twice around the world. Whn am wp? fHSHHHH We are the world's leading oilfield services company. Working 1 globally—often in remote and challenging locations—we invent, design, engineer, manufacture, apply, and maintain technology to help customers find and produce oil and gas safely. Who are we looking for? We offer countless opportunities in the following domains: ■ Engineering, Research, and Operations ^ ■ Geoscience and Petrotechnical ■ Commercial and Business A ^ If you are a self-motivated graduate looking for a dynamic career, apply to join our team. What will you be? careers.slb.com Schlumberger 154 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Proof. It suffices to prove Theorem 3.67 for 1-dimensional Brownian mo¬ tion. So suppose d = 1 and let a < 1/2. Choose p > 1 so large that a < -. 2 p From inequality (3.146) in Proposition 3.66 with X = b(t ) — b(s ) we obtain E (IK*) - mn = E (|6(i — s )f) = C p (E (I b(t - s )| 2 )) p/2 = C p \t-s\ p/2 = C p \t-s\ 1+r , (3.147) where r = p/2 — 1 > pa. An application of Theorem 3.65 yields the desired result. □ The following theorem says that Brownian motion is nowhere differentiable. 3.68. Theorem. Fix a > Then with probability one, t >—> b(t ) is nowhere Holder continuous of order a. More precisely | b(t + h) — b(t)\ P ( inf lim sup h-> 0 \h\ 00 1. Proof. For a proof we refer the reader to the literature; e.g. Simon [[ 121 ], Theorem 5.4. p. 46]. □ In the theory of stochastic integration we will have a need for the following lemma. The following lemma can also be proved by the strong law of large numbers: see e.g. Smythe [123]. 3.69. Lemma. Let {b(s) : s ^ 0} be one-dimensional Brownian motion. Then, P -almost surely, lim n ^oo Yuk=o IM(^ + 1)2 ~ n t) — b(k2~ n t)\ 2 = t. PROOF. Put = \b((k + l)2 -n t) — b (k2~ n t)\ 2 — 2~ n t. Then the variables A^ n , 0 ^ k ^ 2 n — 1, are independent and have expectation 0. So that f 2 n -l \ 2 2 n —1 2 n —i 2 2 A fc , n = JE (A fc , n ) 2 = E (|6 (2 ~ n t) | 2 - 2 ~ n t) k =0 k =0 = 2 n (e | b (2 ~ n t) | 4 - 2E | b (2 ~ n t) | 2 2 ~ n t + 2“ 2n t 2 ) = 2 x 2 ~ n t 2 . (3.148) Tchebychev’s inequality gives °° / p n ~ l \ \ 2 1 2 Hence / P / A k n ) > e < —. Thus we may apply the Borel-Cantelli n =i y \ k =o ) / 6 lemma to prove the claim in Lemma 3.69. □ 3.70. Proposition. Brownian motion is nowhere of bounded variation. 155 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Proof. Just as in the previous lemma we have that, for t > s ^ 0, 2 n —1 lim V 1 |& (s + (k + 1)2 ~ n (t — s)) — b (s + k2~ n (t — s )) 1 2 — (t — s) = 0, n—>oo 1 v v 71 k= 0 P-almost surely. Since Brownian paths are almost surely continuous it follows that (for 6 > 0) 2 n —l 0 < t — s ^ lim V \b (s + (k + l)2 _n (t — s)) — b((s + k2~ n {t — s)) | 2 n —>00 1 v 7 v 71 fc=0 < lim inf max \b (s + (£ + l)2~ n (t — s)) — b((s + £2~ n (t — s))\ n—>co 0s££t£2 n 1 v ' v y| 2 n —1 x Xj \ b ( s + ( fc + — s)) — 6((s + k2~ n (t — s)) | fc=0 ^ sup |£>(ct 2 ) — b(<Ji)\ x variation of b on the interval [s, t]. The statement in the Proposition 3.70 now follows from the continuity of paths. □ Next we will see how to transfer properties of discrete time semi-martingales to continuous time semi-martingales. Most of the results in the remainder of this section are taken from Bhattacharya and Waymire [15]. We begin with an upcrossing inequality for a discrete time sub-martingale. Consider a sequence {Z n : n e N} of real-valued random variables and sigma-fields Ji c J 2 c • • ■, such that, for every n e N, the variable Z n is ^-measurable. An upcrossing of an interval (o, b) by {Z n j is a passage to a value equal to or exceeding b from an value equal to or below a at an earlier time. Define the random variables X n , n e N, by X n = max (Z n — a, 0). If the process { Z n } is a sub-martingale, then so is the process {X n }. The upcrossings of (0, b — a) by { X n } are the upcrossings of the interval (a, b) by { Z n }. We define the successive upcrossing times r/ 2 /,., k e N, of {X n } as follows: r\\ = inf {n ^ 1 : X n = 0}; 772 = inf {n ^ 771 : X n ^ b - a} ; 772^+1 = inf {77 ^ r] 2k : X n = 0}; V 2 k +2 = inf {n ^ 772^+1 :X n >b-a}. Then every 77 k is an {T n }-stopping time. Fix N e N and put 77 , = min ( 77 %, N). Then every 77 is also a stopping time and 77 , = N for k > [7V/2J, the largest integer smaller than or equal to N/2. It follows that X T2k = X N for k > [tt/ 2J and we also have 77 k ^ k and so A’ N 77 . N X. Let U^(a, b) be the number of upcrossings of (a, b) by the process { Z n } at time N. That means U N (a, b) = sup {k^l\r] 2 k^ N} (3.149) with the convention that the supremum over the empty set is 0. Notice that Upf(a, b) is also the number of upcrossings of the interval (0, b — a) by {A",,} in time N. 156 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.71. Proposition (Upcrossing inequality). Let {Z n j be an {3 n }-submartin¬ gale. For each pair (a, b), a < b, the expected number of upcrossings of (a, b ) by Zi ,..., Z N satisfies the inequality: E (U N (a, b)) < E (max (Zjsr — a, 0) — max ( Z\ b — a E (max (Zjv — Z\, 0)) b — a a,0)) (3.150) PROOF. Since X T2k = Xn for k > [A r /2j. we may write (setting To = 1): [JV/2J+1 [JV/2J + 1 X K -X,- 2 (Xr, t _, - x rat _,) + 2 (Xr, t - x nt _,) . (3.151) k =1 k=1 Next let v be the largest integer k with the property that ry. < N, i.e. u is the last time < N of an upcrossing or a downcrossing. It readily follows that U N (a, b ) = [p/ 2J. If v is even, then X T2k — X r , 2k _ 1 ^ b — a provided 2k — 1 < zq X T2k — X T2k _ x = X N — X/v = 0 provided 2k — 1 > v. (3.152) Now suppose that v is odd. Then we have X T2k — X T2k _ 1 5= b — a provided 2k — 1 < u\ X r 2k - X r 2k -1 = x r 2k ~ X v > X r 2k - 0 = X T2k provided 2k - 1 = u; X T2k — X T2k _ x = X N — Xjsf = 0 provided 2k — 1 > u. (3.153) From (3.152) and (3.153) it follows that LAT/2J + 1 [u/2\ 2 (Xr U - x r , k j > y; (x Ta - x Ta _,) k—1 k—1 F [u/2\(b — a) = (b — a){7jv(«, b). (3.154) Consequently [JV/2J+1 Xn — Xi ^ ^ ( X T 2k _i — X r 2k _ 2 ) + (b — a)UN(cL, b). (3.155) k= 1 So far we did not make use of the fact that the process {X n } is a sub-martingale. It then follows that the process {X Tk : k e N} is a { J Tn (-martingale and hence k i—> E (X Tk ) is an increasing sequence of non-negative real numbers. So that (3.155) yields [N/2\ + l E(X N ~X t )» 2 ^(X T2t -X rn _ I ) + (b-a)E(U N (a,b)) k=1 ^ {b — a )E (£//v(a, b )). (3.156) The desired result in Proposition 3.71 follows from (3.156). □ 157 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.72. Theorem. (Sub-martingale convergence theorem) Let { Z n } be a sub-mar- tingale with the property that sup neN E (\Z n \) < oo. Then the sequence {Z n } converges almost surely to an integrable random variable Z<*,. Moreover we have E(|Zoo|) ^ liminf n ^ooE(|Z n |). 3.73. Remark. In general we do not have E(Zoo) = lim^oo E (Z n ). In fact there exist martingales { M n : n e N} such that M n ^ 0, such that E(M n ) = 1, n e N, and such that M x = lim„_ > . x . M n = 0, P-almost surely. To be specific, let {b(s) : s ^ 0} be z/-dimensional Brownian motion starting at x e M !/ and let p(t, x, y ) be the corresponding transition density. Fix t > 0 and y =(= x and put M n p(t/n,b(t - t/n),y) p(t, x, y) (3.157) The process {M n : n e N} defined in (3.157) is P x -martingalc with respect to the sigma-fields T n generated by b(s), 0 < s < t — t/n. / f Maastricht University Join the best at the Maastricht University School of Business and Economics! gjpj* • 33 rd place Financial Times worldwide ranking: MSc International Business • 1 st place: MSc International Business • 1 st place: MSc Financial Economics • 2 nd place: MSc Management of Learning • 2 nd place: MSc Economics • 2 nd place: MSc Econometrics and Operations Research • 2 nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier 'Beste Studies' ranking 2012; Financial Times Global Masters in Management ranking 2012 Maastricht University is the best specialist university in the Netherlands (Elsevier) Master's Open Day: 22 February 2014 www.mastersopenday.nl | 158 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Proof. Let [/(a, b ) be the total number of upcrossings of (a, b ) by the pro¬ cess {Z n : n e N}. Then U]\r(a, b) ] [/(a, b) as N ] go. Therefore, by monotone convergence, E((C/(a, &))= lim E b)) ^ sup ^ ^^ N \) + l a l < ^ (3.158) £V—>oo TVeN b — CL In particular it follows that U(a, b) < go P-almost surely. Hence P (lim inf Z n < a <b < lim sup Z n ) K P(I7(a,6) = go) = 0. (3.159) Since {lim inf Z n < lim sup Z n } = {lim inf Z n < a < b < lim sup Z n } a<b,a,beQ it follows from (3.159) that P (lim inf Z n < lim sup Z n ) = 0. By Fatou’s lemma it follows that E(|Zoo|) = E(liminf n \Z n \) ^ liminf E i\Z n \) < go. This completes the proof of Theorem 3.72. □ 3.74. Corollary. A non-negative martingale { Z n } converges almost surely to a finite limit Z cn . Also E (Zoo) A E (Zi). Remark. Convergence properties for supermartingales {Z n } are obtained from the sub-martingale results applied to {— Z n ). Since a semi-martingale is a difference of two sub-martingales, we also have convergence results for semi¬ martingales. 3.75. Definition. A continuous time process {X(t) : t e M} is called stochasti¬ cally continuous at to if for every e > 0 lim P (|X(f) — X(t 0 )\ > e) = 0. (3.160) 3.76. Remark. Brownian motion possesses almost surely continuous sample paths and is stochastically continuous for every t ^ 0. On the other hand a Poisson process is stochastically continuous, but its sample paths are step functions with unit jumps. In fact, for t > s, X(t) 5= A"(sj P-almost surely and, again for t > s, P(X(f) — X(s) for t > s and for e > 0, n) = e A(t _ 3 ) (Mt ~ s)) n n! and hence, always ¥(X(t)-X(s) ^ e) < e a (t-s) y> fy 5 )) n\ n= 1 1 -e“ A(t “ s) . 3.77. Theorem. Let {X(t) : t ^ 0} be a sub-martingale or a super-martingale that is stochastically continuous at each t ^ 0. Then there exists a process <X(t):t^0> with the following properties: (i) (stochastic equivalence) jx(f)j that P (x(t) = X(t) is equivalent to {X(t )} in the sense for every t ^ 0; 159 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales (ii) (sample path regularity) with probability 1 the sample paths of the pro¬ cess jx(i) : t ^ oj are bounded on compact intervals [a, b], a < b < co, are right-continuous and possess left-hand limits at each t > 0 (in other words \x(t):t^ 0 [ is cadlag). Proof. Fix T > 0 and let Qt denote the set of rational numbers in [0,T]. Write Qt = Un=i where R n is a finite subset of [0, T] and where T e Ri a i ?2 ci R 2 a ■ ■ ■. By Doob’s maximal inequality for sub-martingales we have P fmax |X(f)| > a) < E I X ( T )I n = 12,... \teRn ) A and hence P ( sup |X(t)| > A ) ^ lim P (max |X(t)| > A ) ^ , n = 1,2,... \te Qt ) \teR n 1 ) A For Doob’s maximal inequality see e.g. Proposition 3.107 or Theorem 5.110. In particular, the paths of {X(t) : t e Q r } are bounded with probability 1. Let (c, d) be any interval in M and let U^ T \c, d) denote the number of upcrossings of (c, d) by the process {X(t) : t e Qt}. Then U^(c, d ) is the limit of the number U^(c, d) of upcrossings of (c,d) by {X(t) : t e R n j as n tends to co. By the upcrossing inequality we have E(U {n} (c, d)) E(|X(r)|) + |c d — c (3.161) Since U^(c, d) increases with n it follows from (3.161) that (3.162) and hence that U^ T fc, d) is almost surely finite. Taking unions over all intervals (c,d), with c, d e Q, and c < d, it follows with probability 1 that the process {X(t) : t e Qt} has only finitely many upcrossings of any interval. In particular, therefore, left- and right-hand limits must exist at each t < T P-almost surely. To construct a right-continuous version of {X(t)} we define jw(t) : t ^ oj as follows: X(t) = lim s p )S6 Q X(s) for t <T. That this process jx(f) j is stochas¬ tically equivalent to (W(f)} follows from the stochastic continuity of the process (X(f)}. Further details are left to the reader. This completes the proof of Theorem 3.77. □ Next we prove Doob’s optional sampling for continuous time sub-martingales (that are right-continuous) and a similar result holds for martingales (where the inequality sign in (3.163) is replaced with an equality) and for super-martingales (where the inequality is reversed). For discrete sub-martingales the result will be taken for granted: see Theorems 5.104 and 5.114. 160 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.78. Theorem. Let { X(t ) : t ^ 0} be a right-continuous sub-martingale of class (DL) and let T be a stopping time. Suppose t ^ s. Then E [X (min(f, T)) | fF s ] ^ X(min(s, T)), P -almost surely. (3.163) Proof. Put s n = 2~ n \2 n s], t n = 2~ n \2 n t] and T n = 2~ n \2 n T], If A belongs to ‘J s , then A also belongs to 3 r Sn for all n e N. From Doob’s optional sampling for discrete time sub-martingales we infer, upon using the (DL)-property, E [X (min(f n , T n )) 1 A ] ^E[X (min(.s n , T n )) 1 A ]. (3.164) Upon letting n tend to oo and using the right-continuity of the process t > X(t), t ^ 0, we infer E [X (min(f, T )) 1^] ^ E [X (min(s, T )) 1^], (3.165) where A e T s is arbitrary. Consequently the result in (3.163) follows from (3.165), and so the proof of Theorem is complete 3.78. □ > Apply now REDEFINE YOUR FUTURE AXA GLOBAL GRADUATE PROGRAM 2015 redefining /standards Qr 161 ^0 Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 6. Stochastic integrals, Ito’s formula The assumptions are as in Section 4. The process {b(t) : t ^ 0} is assumed to be one-dimensional Brownian motion and hence the process t i—> b(t) 2 — t is a martingale: see Proposition 3.23. The following proposition contains the basic ingredients of (the definition of) a stochastic integral. 3.79. Proposition. Let si,..., s n and be non-negative numbers for which Sj -1 < tj -1 < Sj < tj, 2 < j < n. Let fi,..., f n be bounded random variables which are measurable with respect to £F Sl ,..., 3 r Sn respectively. Put Y(s, u>) = 2J =1 /i(^) 1 (^,t j ]( s ) and write f Y ( s > -) db ( s ) = 2" =1 fj tj)) - b(mm(t, Sj ))} . Jo J The following assertions hold true: (a) (b) The process |^F(s)d6(s) : t ^ o| is a martingale and the process Y (s)db(s) S j : t ^ o| is a submartingale. The process | (So ^ ( s )db(s)j — Y ( s) 2 ds : t ^ o| is a martingale, (ltd isometry) The following equality is valid: E f Y(s)db(i Jo = E f Y(s) 2 ds Jo (3.166) The equality in assertion (c) is called the Ito isometry. It is an extremely important equality: the entire Ito calculus is justified by the use of the equality in (3.166). PROOF. The more or less straightforward calculations are left as an exercise to the reader. We insert some ways to simplify the computations. Let F and G be predictable processes of the form F(s) = /l( u ,oo)( s ) an d G(s) = gl( v ^ <Xl ' ) (s), where / is measurable for the a-fie Id T u and g for T„. Put h(F) = f F(s)db(s) := f (b(t) -b(mm(u,t))) Jo and similarly write h(G) = f G(s)db(s) = g ( b(t) - b( min(u, t))). Jo Without loss of generality we assume v ^ u (otherwise we interchange the role of F and G). We begin with a proof of (a). Upon employing linearity it suffices to show that the process t <—>■ L(F) is a martingale. (Also notice that 162 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales So /I (u,v](s)db(s) = I t (F) - where 7\(s) = /l(„ i00) (s).) Fix t > s ^ 0 and consider E (. I t (F ) | F s ) - I S (F ) = E (. I t (F ) - I S (F) \ F s ) = E (/ ( b{t) — &(min(w, t )) — b(s ) + &(min(w, s))) | IF S ) = E (E (/ (b(t) - b(mm(u, t )) - b(s) + b(mm(u, s))) | lBnm(max(«, s ),t)) | 3^) = E (/E ((6(i) - 6(min(u, t)) - b(s ) + 6(min(u, s))) | 3 r mm( ma x(u, a ),t0 | 3\ s ) (Brownian motion is a martingale) = E (/ ((6(min(max(n, s), t)) — b(mm(u, t)) — b(s) + &(min(w, s)))) | 3^) = 0, proving that the process t >—> I t (F ) is a martingale indeed. Next we shall prove that the process t >—> I t (F)I t (G ) — F(t)G(t)cIt is a martingale. Using bilinearity in F and G yields a proof of (b) and hence also of (c). Again we fix t > s and consider E (l t (F)I t (G) - £ F(r)G(r)dr | ?£ - (l a (F)I a (G) - £ F(r)G(T)d£) = E (l t (F)I t (G) - £ F(r)G(r)dr - (l s {F)I s (G) - £ F(r)G(r)dr^ | = E £/ t (F) - I a (F)) (. I t (G ) - I a (G)) - £ F(r)G(r)dr | + E (l a (F) ( I t (G ) - 7 S (G)) + (7 t (F) - 7 S (F)) I S (G) | J s ) = E £/ t (F) - J S (F)) (7 t (G) - I a (G)) - £ F(r)G(r)dr | ?£ + I a (F )E (7 t (G) - 7 S (G) | 5 a ) + E (l t (F) - I a (F ) | 5B) 7 fl (G) (use the martingale property of 7 f (F) and I t (G )) = E (I,(F) - /.(F)) (/,(G) - /.(G)) - f F(t)G(t) dr J s -E[/g(i>(t)-i> (min(max(ti, s),t))) (b(t) — &(min(max(u, s ), *))) -fg (t - min(max(n, v, s),t)) | 3%*] (use v ^ u and put u a> t = min(max(n, s),t), v s j = min(max(u, s),t)) E fd ((W) ~ b ( v s,t)Y +fg (b(v a>t ) - b(u a>t )) (b(t) - b(v a>t )) - fg (t - min(max(u, v, s), t)) | 5B)] = E fa - b(v 8j t)) - (t - v S: t )) | IB + E [(fg (b(v a , t ) - b(u a , t )) (i b(t ) - b(v att ))) | J a ] = E /0E - &(w s ,t)) 2 -(t- v a M | J, ' V s ,t 3B 163 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales + E [fg (b(v a>t ) - b(u 8tt )) x E [(b(t) - b(v a>t )) | | J s ] (the processes {b(s)} and {b(s) 2 — s} are martingales) = E (fg .0 | 5 a ) + E (fg (b(v a , t ) - b(u a , t )) .0 | T s ) = 0. The latter yields a proof of (b) (via bilinearity). Altogether this finishes the proof of Proposition 3.79. □ 3.80. Definition. A process of the form F(s,u) = Y? j=1 fj( u ) 1 (s j ,t j ](s), where 0 < Sj-i < t,_i < Sj < tj, 2 ^ j ^ n, and where the functions fi,... ,f n are bounded and measurable with respect to T Sl ,..., 3 r Sn respectively is called a simple predictable process. Empowering People. Improving Business. Norwegian Business School is one of Europe's largest business schools welcoming more than 20,000 students. Our programmes provide a stimulating and multi-cultural learning environment with an international outlook ultimately providing students with professional skills to meet the increasing needs of businesses. B! offers four different two-yea i; full-time Master of Science (MSc) programmes that are taught entirely in English and have been designed to provide professional skills to meet the increasing need of businesses.The MSc programmes provide a stimulating and multi¬ cultural learning environment to give you the best platform to launch into your career * MSc in Business * MSc in Financial Economics * MSc in Strategic Marketing Management NORWEGIAN BUSINESS SCHOOL EFMD EQUIS *ffi * MSc in Leadership and Organisational Psychology www.bi.edu/master 164 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.81. Definition. Again let b be Brownian motion with drift zero and let n 2 (6) be the vector space of all predictable processes F with the property that |F|; := E rCC IW| S Jo ds 1 <oo. Let Q be the cr-additive measure, defined on the predictable field II, determined by Q (A x (s, t]) = E [ 1 ^ 4 ] (t — s) = P(yl)(t — s), A e T s . (3.167) The measure Q is called the Doleans measure for Brownian motion. Then it follows that n 2 (&) = L 2 ([0, 00) x fl, n, Q). Moreover we have Uplift = J \F\ 2 dQ, FeU 2 (b). It also follows that, for given F e n 2 (6), there exists a sequence of simple processes (F n : n e N), which are predictable, such that lim n ^oo \\F n — F\\ b = 0. Hence in view of Proposition 3.79 it is obvious how to define F(s) db(s), t ^ 0, for F e n 2 (6). In fact f F( y s)db(s ) = L 2 - lim f F n (s)db(s ), JO n^oo J 0 where the sequence (F n : n e N) verifies lim n ^oo \\F n F\\ b = 0 and where F n belongs to n 2 (&). Let n 3 (fe) be the vector space of all predictable processes F for which the integrals ^ |T(s)| 2 ds are finite P-almost surely for all t > 0. In order to extend the definition of stochastic integral to processes F 6 n 3 (6) we proceed as follows. Define the stopping times T n , n e N, in the following fashion: T„ = inf < t > 0 : f |F(s)| 2 ds > n Jo (3.168) We also write F n (s) = F(s)l{T n > s } an d we observe that F n is a predictable process with J |F n (s)| 2 ds F n. Moreover it follows that for n > m the expression f F n (s)db(s)~ f F m (s)db(s) = f F(s)l (Tmjmin(Tii;t) ](s)d6(s) (3.169) Jo Jo J vanishes almost everywhere on the event {T m > t}. So it makes sense to write f F(s)db(s) = f F m (s)db(s), on { T m > t}. Jo Jo Since lim n ^.ooT n = 00, P-almost surely, the quantity ^ 0 F(s)db(s) is unambigu¬ ously defined. Hence the integral $ F(s)db(s) is well defined for processes F belonging to n 3 (&). 3.82. Corollary. Let b be Brownian motion and let F and G be processes in n 3 (&). The following processes are local martingales: "t J F(s)db(s) : t > 0 j , j J G(s)db(s ) : t ^ 0 (3.170) 165 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales J F(s)db(s^j - J |F(s)| 2 ds:t> 0 J ; (3.171) f F(s)db(s ) f G(s)db(s) - f F(s)G(s)ds : t> ol. (3.172) Jo Jo Jo J Put X(t) = F(s)db(s) and Y(t ) = \ ) t 0 G(s)db(s). The following identity is valid: X(t)Y(t ) - f F(s)G(s)ds = f F(s)Y(s)db(s) + f X(s)G(s)db(s). (3.173) Jo Jo Jo Proof. The assertions (3.170), (3.171) and (3.172) follow from Proposition 3.79 together with taking appropriate limits. For the proof of (3.173) we first take F = G = 1. Then (3.173) reduces to showing that b(t) 2 -2 f b(s)db(s) - t = 0. (3.174) Jo Notice that (3.174) is equivalent to 2 §* b(s)db(s) = b(t) 2 — t, t ^ 0. For the proof of (3.174) we use Lemma 3.69. to conclude: b(t) 2 — 2 f b(s)db(s ) Jo / 2 n — = lim b(t) 2 - 2 J] b (k2~ n t) (b ((k + l)2“ n f) - b (k2~ n t) - t) — t 2 n —1 k =0 1™ (s' (ft ((fc + 1)2—*) 2 - 6 (fc2-“0 : \ k =0 2 n —1 -2 J] b (k2~ n t) (b ((k + l)2 _n t) - b (k2~ n )) - t k =0 /2 n -l lim J (b((k + \)2-H)-b(k2-’'t))- t = 0. \ k =0 For the proof of (3.173) we then take F(s) = /l( Sl)00 )(s) and G(s) = g 1( S2)00 ), where / is bounded and measurable with respect to T Sl and g is measurable with respect to T S2 . Formula (3.174) will then yield the desired result. Then we pass over to linear combinations and finally to limits. This completes the proof of Corollary 3.82. □ 3.83. Proposition. Stochastic integrals with integrands inU 3 (b ) are continuous F-almost surely. PROOF. It suffices to prove the result for integrands in n 2 (6). Since Brow¬ nian motion is almost surely continuous, it follows that stochastic integrals of simple predictable processes are continuous. Let F be in n 2 (6) and choose a 166 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales sequence (. F n ) ne ^ of simple predictable processes with the property that -to \ \ 1/2 / / rto \\ !/2 rto m s ) - F n (s)\ ds lim [ E J^°|F n ^(s)-F n (s)| 2 ds^ \F n+i (s) - F n+e -i(s )| 2 ds 1/2 oo E 2 £= 1 — n —£—2 2“ n_1 From Proposition 3.63 it follows that, for k e N, E sup O^t^to r <: 2 E <=i V 00 / 2 2 E £=1 00 (■ F n+k (s ) - F n (s))db(s) f (F n+ ^(s) - F n+ ^_i(s)) d&(s) Jo f ( F n+i (s ) - F n+ /_i(s)) d6(s) Jo sup O^t^to rt 0 - 2 E( e [ r=i ^ LJo |F n+ ^(s) - -F n+ ^_i(s)| 2 ds < 2 “ (3.175) From (3.175) the sample path continuity of stochastic integrals immediately follows. This completes the proof of Proposition 3.83. □ 167 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.84. Remark. The theory in this section can be extended to (continuous) martingales instead of Brownian motion. To be precise, let t >-*■ M(t) be a continuous martingale with quadratic variation process t >—> (M, M) (t). Then the process t >—► M(t ) 2 — (M, M) (t) is a martingale, and the space II 2 (&) should be replaced with n 2 (M), the space of all predictable processes t >—► F(t) with the property that IITll = E \F(s)\ 2 d(M,M) («) < 00 . (3.176) The corresponding Doleans measure Qm is given by Qm (A x (s, f]) = E [1 A «M, M) {t) - (M, M) (s))], AeJ s , s<t. (3.177) It follows that n 2 (m) = L 2 (n x [o,oo),n,g M ). The space Il 3 (M) consists of those predictable processes t <—>■ F(t ) which have the property that \F(s)\ 2 d (M, M) s are finite P-almost surely for all t > 0. The definition of stochastic integral to processes F e II 3 (M) we proceed as follows. Define the stopping times T n , n e M, in the following fashion: T n = inf |t > 0 : J |F(s)| 2 d (M, M) (s) > n j . (3.178) As in the case of Brownian motion these stopping times can be used to define stochastic integrals of the form F(s) dM(s), F e n 3 (M). These integrals are then local martingales. Next we extend the equality in (3.173) to the multi-dimensional situation. 3.85. Proposition. Let s >-*■ a(s) = (c r jfc(s)) 1<j fc<J/ be a matrix with predictable entries and with the property that the expression rt " rt Z[ j,k= 1 E|cr ifc (s)| ds (3.179) is finite for every t > 0. Put a,ij(s) = Yjk= i a ik{ s ) CF jk{s), 1 < i, j < n. Further¬ more let {b(s) = (bi(s),... ,b u (s)) : s ^ 0} be u-dimensional Brownian motion. Put Mj(t ) = X!fc=i )o a jk{ s )dbk(s), 1 < j < v. Then the following identity is valid: (3.180) V rt V rt rt I Afj(s)<jj;j(s)d6^(s) + I (s)ATj(s)c?6/j(s) + I Ujj(s)ds. k= l k= l Proof. First we suppose v = 2, Mfit) = bi(t) and M 2 (t) (3.180) reads as follows: bi(t)b 2 (t) = f b 1 (s)db 2 (s) + f b 2 (s)dbi(s). Jo Jo b 2 (t). Then (3.181) 168 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In order to prove (3.181) we write bi(t)b 2 (t)~ f 6 1 (s)d 6 2 (s) - f 6 2 (s)d 6 1 (s) Jo Jo 2 n —1 f = lim | h ((k + 1 ) 2 ~ n t) b 2 ((k + 1 ) 2 ~ n t) - b x ( k2~ n t ) 6 2 ( k2~ n t ) n ^°° fc =0 ( - 61 (fc 2 “ n t) (ft 2 ((k + 1)2 ~ n t) - b 2 ( k2~ n t )) - b 2 (k2~ n t) (b x ((fc + l) 2 _n t) - b\ (k2~ n t)) 2 n —1 Jim 2 ( 6 i ((fc + 1)2 ~ n t) - h (k2~ n t)) (b 2 ((k + 1)2 ~ n i) - b 2 (k2~ n t)) . (3.182) fc =0 The limit in (3.182) vanishes, because by independence and martingale proper¬ ties of the processes b\ and b 2 , we infer 2 n —1 E Yi ( & i (( fc + 1 ) 2 ~ nt ) - {k2~ n t)) (b 2 ((k + 1)2 ~ n t) - b 2 k =0 ( 2 n —1 2 (6i ((fc + l)2 _n t) - l h (k2~ n t)) 2 (b 2 ((k + l)2 _n t) k= 0 2 n —1 = E [b x ((fc + l)2 _n t) - b x (k2~ n t)f E (6 2 ((fc + l)2 -n t) k= 0 2 n —1 = Y i 2 ~ n t) 2 = 2~ n t 2 . k =0 (k2~ n t)) — b 2 (fc 2 ""*)) 2 j — b 2 (k2~ n t)) 2 (3.183) From Borel-Cantelli’s lemma it then easily follows that the limit in (3.182) van¬ ishes and hence that equality (3.181) is true. The validity of (3.180) is then checked for the special case that crj k (s) = fjk^( Sjk ,co){s), where fj k is measurable with respect to “J s . k . The general statement follows via bi-linearity and a lim¬ iting procedure together with equality (3.173) in Corollary 5.142. The proof of Proposition 3.85 is now complete. □ Next let M(t) = (M x (t ),..., M u (t)) be a i/-dimensional martingale as in Propo¬ sition 3.85 and let A(t) = (A x (t ),..., A u (t)) be an adapted z^-dimensional pro¬ cess that P-almost surely is of bounded variation on [0, t] for every t > 0. This means that sup sup |A(sj) — A(sj_i)| is finite P- almost surely for allt > 0. ne N 0^so<si<---s n ^t It follows that the random set function /j, A : (a, b] > A(b) — A (a) extends to an PC-valued measure on [ 0 , t] for every t > 0 . Stieltjes integrals of the form F(s)dA(s) may be interpreted as ^F(s)dA(s) = F(s)dju A (s). The process A may have jumps. This is not the case for the process M. The latter 169 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales follows from Proposition 3.83. The process X : = A + M is a ^-dimensional semi-martingale with the property that E (|M(i)| 2 ) < go, t > 0. Put Mt) = E»« (*(») - ,*(<<-)) = 2 (M(s) - M(s—)) + 2 (A(s) - A(s-)) = J A (t). S^t s^t The definition of J 4 (t) does not pose to much of a problem. In fact for P-almost all u) the sum ^ |A(s,o;) — t4(s—, w)| < oo. The process {X(t) — Jxif) • t ^ 0} is P-almost surely continuous. Brain power By 2020, wind could provide one-tenth of our planet’s electricity needs. Already today, SKF’s innovative know¬ how is crucial to running a large proportion of the world’s wind turbines. Up to 25 7o of the generating costs relate to mainte¬ nance. These can be reduced dramatically thanks to our (^sterns for on-line condition monitoring and automatic lul|kation. We help make it more economical to create cleanSkdneaper energy out of thin air. By sh?fe|ig our experience, expertise, and creativity, industries can boost performance beyond expectations. Therefore we need the best employees who can kneet this challenge! Power of Knowledge Engineering Plug into The Power of Knowledge Engineering. Visit us at www.skf.com/knowledge 170 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales The following result is the fundamental theorem in stochastic calculus. 3.86. Theorem (Ito’s formula). Let X = (Xi,...,X u ) = A + M be a u- dimensional local semi-martingale as described above, and let f : IF —> R be a twice continuously differentiable function. Put atj ft) = 'ffjk=i <T ik(t) cr jk(t), 1 < i, j < v. Then, P -almost surely, - f(X( 0)) + 2(/(*(»)) - / (X(s—)) - V/ (X(s- j). (X(s) - X(s-)) S^t ' + f V/(X(s-)) • dX(s) + \ V C D i D j f(X(s))a ij (s)ds. (3.184) Jo * iJ=1 Jo Before we prove Theorem 3.86 we want to make some comments and we want to give a reformulation of Ito’s formula. Moreover, we shall not prove Ito’s formula in its full generality. We shall content ourselves with a proof with A = 0 . Remark. The integral ^ Vf(X(s—)) dX(s) has the interpretation: ( Vf(X(s-)) dX(s) Jo = 2 ( \ Dif(X(s-)) dMfs) + f Dif(X( S -)) dAi(s) i=1 \Jo Jo = y ( f Dif(X(s-)) dMfs) + f Dif(X( S -)) dAfs) i=1 \J0 Jo (3.185) 2(2 fn i f(X(s-))a ik (s)db k (s)+ f A i= 1 \fc=l *^° f(X(s-))dA t (s) . Here X = M + A is the decomposition of the semi-martingale in a martingale part M and a process A which is locally of bounded variation. For z/-dimensional Brownian motion we have the following corollary. 3.87. Corollary. Let b(t) = (bft ),..., Kft)) be is-dimensional Brownian mo¬ tion. Let f : IF —*• M be a twice continuously differentiable function. Then, P -almost surely, m*)) = HHP)) + f Xf(b(s))db(s) + 1 f A f(b(s))ds. (3.186) In fact it suffices to suppose that the functions Dif ,..., D u f and Dff ,..., Dff are continuous. Next we reformulate Ito’s formula. 3.88. Theorem. Let X = (Xi,...,X„) be a is-dimensional right continuous semi-martingale as in Theorem 3.86 and let f : PC —» M be a twice continuously differentiable function. Then, P -almost surely, f(X(t)) = f(X(0 )) + f V/ (X( S -)) • dX(s) (3.187) JO 171 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales + Y f \\l-a)D i D J f((l-a)X(s-)+aX(s))dad[X i ,X j \(s), A A -1 JO JO i j—l 'JO JO where [X„ Xj] (t) = f a tj ( S )d S + ^ (A',( s ) - X,(s-)) (X,( s ) - X,(s-)) . (3.188) Jo 8^t Remark. In the proof below we employ the following notation. Let Mi be martingale of the form v rt Mi(t) := 2 a ik (s)db k (s). k= i J o Then quadratic covariation process (Mj, Mj) (t) satisfies V r-t (Mi, Mj) (0 = 2 a ik (s)a jk (s) ds. k =l Proof of Theorems 3.88 and 3.86. Since, for a and b in f(b) -/(a) = V/(a).(6 — a) V r i (3.189) IS pi + 2 0 “ cr )DiDjf ((1 - a)a + ab) da x - a^bj - a,) i,j =1 and since [V, X,] (t) = f aii ( S )d S + 2 (XiM - *<(«-)) (Xj( s ) - X,( S -)), J° it follows that z' r t r i 2 f f (l-a)D i D J f((l-a)X(s-) + aX(s))dad[X l ,X j ](s) ij =i Jo Jo = 2 f f (! ** v)DiDjf ((1 ^a)X(s-) + aX(s)) daa i:j (s)ds i,j =l J o Jo + 2 2 fo - °)DiDjf ((1 - a)X(s~) + aX(s)) da i,j = l s^t Jo x (X<(s) ~ X<(s-)) (Xj(s) - X^s-)) - y f f (l-cr)D i D J f((l-c7)X(s-) + aX(s))daa ij (s)ds ij-i J « J » + 2 t/(X(s)) - f (X( S -)) - V/ (X(s-)). (X(s) - X(s-m . (3.190) So the formulas in Theorem 3.88 and Theorem 3.86 are equivalent. Also notice that, since \ ) t Q aij(s)ds is a continuous process of finite variation (locally), we 172 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales have JT< (1 — a)DiDjf ((1 — (j)X(s—) + crA(s)) daaij(s)ds J7< rt (1 — a)DiDjf ((1 — a)X(s-) + aX(s —)) daaij(s)ds = \^D,D J f(X(s-))a, j (s)ds. Hence it suffices to prove equality (3.187) in Theorem 3.88. Assume A(0) = M( 0) and hence A(0) = 0. Upon stopping we may and do assume that in X = M + A, \X(t —)| < L and var A(t—) < L. This can be achieved by replacing X(t) with A (min(f, r)), where r is the stopping time defined by r = inf {s > 0 : max (\M(s) \, varA(s)) > L} . Here varA(s) is defined by var A(s) = sup ^ |A(sj) - A(s i _i)| : 0 < s 0 < s 1 < ... < s n = s j- . 3 = 1 Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Next we define, for every n e N, the sequence of stopping times {T nk : k e N} as follows: T n .o — 0; T nM i = inf is > T n , k : max(s - T n>fc , |X(s) - X(T n>k ) |) > - ( n Since max(T n;fc+1 — T Ujk , \X(T Utk+ i ) — X(T Hjk )|) ^ , Th it follows that lim T n & = go, P- almost surely. Moreover, since fc—► oo max (T n k+ i - T n>k , \X(T n>k+1 ~) - X(T„ )fc )|) < n (3.191) we have T nk+ i — T n k ^ . Next we write: 1 n f(X(t )) - f(X( 0 )) 00 = 2 {/ (X (T n>k+1 a t-)) - / (X (T n , fc a t)) fc =0 +/(* (T n ,/c+l a t))-f(X(T n ,/c+l ^ *-))} 00 f r Tn ’ /e+iAt_ = 2l Vf(X(T n , k At))-dX(s) k =0 f i/ + 2 ( X - a ) D i D jf ((! - <0^ (■ T n,k At) +aX (T nM 1 a t-)) dcr *,l=i x (V Un,fc+i A ) — Xi {T n , k a t)) (Xj {T n ,k +1 A ) — Xj (T n> fe a t)) + / (X (T n , fc+1 At))- / (X (T nM1 a H)|- (3-192) On the other hand we also have: f Vf(X(s-))dX(s) jo + y f [\l - a)D t D 3 f ((I - a)X{s~) + aX(s)) dadiX^X^is) i,j =l J o do 00 f r T n,fe+i At - = 2] Vf(X(s-))-dX{s) k=0 f dT u , fc At + 2 f "’ fc+1 * f (l - ^DtDjf ((1 - a)X(s-) + aX(s)) dadiX^X,]^) ij=oJT n , k At JO + V/ (X (T n;fc+ i a t —)). (X (T n;fc+ i At) — X (T njfc+ i a t—)) + 2 0 - a ) D i D jf ((! - cr ) x ( T n,k+1 A t~) + aX (T nM 1 A t)) dcr i,. 7 = 1 174 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales x (Xi (T nj k+ 1 At) — Xi (T n? / C+ 1 a t—)) (Xj (T n?fc+1 At) — Xj (T n> jfe + 1 a t —)) 00 ( r T ^+ lAt- su Xf(X(s-))-dX(s) k =0 V ln ’ k T n b a t + Y r"’ fc+lAt f(l - ((1 - a)X(s-) + <7l( S )) rfarf [I t) I,] ( 5 ) i,j=0 ^ T n,k At JO + / A (Afc+l At))- / (X (Tn.fc+l A t-))|. (3.193) Upon subtracting (3.193) from (3.192) we infer by employing Proposition 3.85: - f(X( 0)) - f V/ (A(o-)) <fA(s) Jo - y f f (1 - a) A^i/ ((1 - W [li, I 3 ] (s) i,j= i J o Jo oo f r>i - 2-1 fc=0 l J1 1 r * r ^'n,k +1 A t (V/ (A(s-)) - V/ (A' (T n , t ))) ■ dX(s) ' T n Y f { f (f - <x) DiDjf ((1 - a)X (T, hk At) + aX ( T nMl a t-)) da ij=\^T n k At V Jo - J (1 - cr)DiDjf ((1 - (r)X(s-) + aX(s)) da | d [Xj, Xj] (s) V + Y 0 - cr) DiDjf ((1 - a)X (T Ujk At) + aX (T nM1 a t-)) da i,j=1 X |(Xj (T n> k+1 A t—) — Xi (Tn )k A t)) (Xj (T n ,fc + 1 A t —) — Xj (T n> fc A t)) — [Xj, Xj] (T n ,fc +1 a t —) + [Xj, Xj] (T n> fc a f) 00 f rT„, fc+ 1 At- s -r k = 0 I, JT n,k^t (V/ (X(s-)) - V/ (X (r„, fc ))) • dX( S ) + S f "’ fe+1 * f (1-0-) {DiDjf ((1 - a)X(T U}k a t)+aX (T n>k+1 a t-)) JT ri ' k At Jo -DiDjf ((1 - a)X(s-) + crX(s))} dad [Xj, X,] (s) V r* 1 + 2 (1 - a)DiDjf ((1 - a)X(T Utk a t) + aX (T n ,fc+i a t-)) da M = 1 rT n ,k+1 x < (Xj(s—) — Xj (T n>k a t)) dXj(s) JT njk At 175 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales rT n ,k + l A t + {Xj{s-)-Xj{T n , k ^t))dXi{s) (3.194) We shall estimate the following quantities: 00 rTn,k + l^t — ^ r-Ln,k +1 E I I 2 (A/ (X(s-)) - A/ {X (T n , fc a t))) ■ dMi(s) 1 | ; (3.195) k=o ^ At ^ rT n<k+ 1 At- \ 2 (A/ (X(s-)) - DJ (X (T n , k a t))) • dAfs) ; (3.196) k = 0^Tn,k A t ) ( 00 rT njk +l A t- pi 2 (1 - <?) {D l D j f ((1 - a)X(T n , k a t) + aX (T n , fc+1 a t-)) \ k =0 % ' T n,k A t Jo (3.197) E E -DiDjf ((1 - a)X(s-) + aX(s))} dad [X t , X,] (s) E ( 00 pi 2 (1 - a)DiDjf ((1 - a)X(T n)k a t) + aX (T nMl a £-)) da k=0 rTn,k+l^t— \ 2 \ X (Xi(s-)r- Xi(T ntk At))dMj(s)\ ; (3.198) JT n , k At ) ) ( 00 y (1- (j)DiDjf ((1 - a)X(T nt k a t)+aX ( T Ujk+1 a t-)) da k=0 J o (Xj(s—) — Xi (T U)k a 0) dAj(s) JT n ,k A t X (3.199) Since the process (A/ (X(s—)) — A/ (X (T nk a £))) dMfs), u ^ 0, is a mar¬ tingale, the quantity in (3.195) verifies , / 00 fWfc + lAt- X 2N Ely (A/ (X(s-)) - A/ (X (T n , fc A t))) • dMfs) \ k =odT ntk At w I ( rT n ,k+i^t- 2 E (A/ (X(s-)) - DJ (X (T n , fc a t))) • dMi(s) £■=-0 \ \JTnUAt k = 0 \ \ Kjl n,k 00 / ^T n>fc + 1 At- S E 1 _ (A/ (X(s-)) - A/ (X (T n , fc A £)))" • d {Mi) (s) fc = 0 \JTnjAt < sup |A/(2/) - D i f(x)\\E((M i ){t)). (3.200) x,yeR. u :\y— x|^l/n,max(|2c|,|2/|)<2L Similarly we obtain an estimate for the quantity in (3.198): 00 pi E| I y J (1 - a)DiDjf ((1 - a)X{T n)k a t) + aX {T nM1 a £-)) da 176 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales rT n , k +1 A t 0 - °) D i D jf ((1 - cr)X(T njk a t) +crX ( T nM1 a /-)) da f*T n ,k +1 JT n ^/\t ( 00 r^n,A: + l A ^ — sup |A£>i/(y)|E 2 \fc=0 'JTn,k At (3.201) E < 1 |j/|<2L 2 U.|sC2L sup lAA'/MIx^EKMiXt)). qaiteye Challenge the way we run EXPERIENCE THE POWER OF FULL ENGAGEMENT... RUN FASTER. — p RUN LONGER.. RUN EASIER... > 177 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales The other estimates are even easier: = E ^ 00 rT n ,k+i*t— 2 (Dif (X(s-)) - Dif (X (T„, fc ))) • dAi(s) k — 0 ,k f(y ) - Dif(x) I .E Q IdAis)^ , (3.202) < sup |D x,yeMy:\y— a:|^l/n,max(|a;|,|y|)^2L 00 rT n . k+1 At- rl E ^ r*n,k+ lAt- r 1 2 (1 - a) {DiDjf ((1 - a)X(T nifc Af) + al (T n>fc+ i a £-)) Jo -DiDjf ((1 - a)X(s-) + aX(s))} dad [X h X)] (s) sup |Ar)j/(y) - DiDjf (x)\ ^ x,yeR u :\y— a:|^l/n,max(|x|,|y|)^2L xe([ Idp^lMl) sup |A^-/(y) - A^j/(®)| ^ x,yeM^:|y—x|^l/n,max(|a:|,|2/|)<2Z/ X y/E([Xi,Xi] (0)^E([X„X i ] (0), and (3.203) E 00 r i V (1 - a)DiDjf ((1 - a)X(T U:k a t) + aX (: T n>k+1 a t-)) da k =o rTn,k+l (Xi(s-) - Xi (T ntk A t)) dAj(s) X 1 < — sup | DiDj yeR l ',|j/|< 2 L f(y)\ E (J o I dAj(s) (3.204) The inequality (3.203) will be established shortly. The quantities (3.200), (3.201), (3.202), (3.203) and (3.204) tend to zero if n tends to infinity. Conse¬ quently, from (3.194) it then follows that, P-almost surely, }(X(t)) = /(X(0)) + f Vf(X(s))dX(s) (3.205) Jo + Y f \\l-a)D i D j f((l^a)X(s-)+aX(s))dad[X i ,X j ](s). i,j = l J ° J ° So that the formula of Ito has been established now. For completeness we prove the inequality E £ Id [V, x,} Ml) ' v'E([X.,X 1 ](t))y , E([M,M](t)). (3.206) 178 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales A proof of (3.206) will establish (3.203). For an appropriate sequence of subdivi¬ sions 0 = Sq 1 ^ < ^ < • • • < s^ n = t we have with, temporarily, \Xj\ = [X^ X^], rt \dlXuXj] (s)| = lim £ \lX it Xj] ( 4 ”>) - lX lt Xj] ( 4 ”_\ Jo n ^ 00 ^i l V 7 V <4 Jim 2 J[Xi] (4"’) - [Xi] (4 %)J[Xj] (4”’) - (44 % k =1 V ’ / N n \ 1 / 2 « Um (2 ([*<,*] (4">) - [A,,*] (4"_>i)) j ( £ (iXi.Xj] (4” 1 ) - [Xj,Xj] (4” , 1 ))J “(PC, A',] (t) - [A,, A,] ( 0)) 1/2 ([Aj, Aj] (t) - [A,, Xj] ( 0)) 1/2 (3.207) Taking expectations and using the inequality of Cauchy-Schwartz once more yields the desired result. This completes the proofs of Theorems 3.86 and 3.88. □ Remark. In the proof of equality (3.194) there is a gap. It is correct if the process ^4 = 0. In order to make the proof complete, Proposition 3.85 has to be supplemented with equalities of the form (M*(f) = Yjk =l So a ik( s )db k (s )): rt v ) = Mi(s)dAj(s ) + V a ik (s)Aj(s)db k (s)-, Jo k=l Jo Ai(t)Aj(t) — f Ai(s)dAj(s ) + f Aj(s)dA t (s). Jo Jo This kind of equalities is true for continuous processes. If jumps are present even more care has to be taken. We continue with some examples. We begin with the heat equation. Example 1. (Heat equation) Let U be an open subset of IT, let f : U —* M be a function in G'o ( E ) and let a : [0, co) x U —*• M be a solution to the following problem: f ^ | A u in [0, co) x U; I u is continuous on [0, oo) x U and u(0, x) = f(x). Moreover we assume that lim u(t,x ) = 0 if b belongs to dU. Then u(t,x ) = x^>b,xEU E x [f(b(t)) : t > t\, where r is the exit time of U: r= inf {s > 0 : b(s) e Of course {b(s) : s ^ 0} stands for //-dimensional Brownian motion. In order to prove this claim we fix t > 0 and we consider the process {M(s) : 0 ^ s < t} defined by M(s) = u(t — s, &(s))l{ T>s }. An application of Ito’s formula yields 179 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales the following identities: r du r M(s)-M( 0) = — I ~ r,b(r))l {T>r} dr + I X7u(t - r, 6(r))l {T>r} • db(r) i r -J Au(t - r,b(r))l {T>r} dr I'i + r - r, b(r )) + u(t - r, b(r )) \ 1 {T>r} dr f Vu(t - r,b(r))l {T>r y db(r) Jo Vu(t - r,b(r)) l {T>r} • db(r). Consequently, the process {M(s) : 0 < s ^ t} is a martingale. It follows that u(t,x) = E x (u(t,b(0))) = E a; (M(0)) = E x (M(t)) = E x (u(0,b(t))l { T >t l) = Ex (/(&(*))l(r>«)) ■ This e-book is made with SetaPDF SETASIGN PDF components for PHP developers www.setasign.com 180 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Example 2 . Let U be an open subset of M l/ , let / belong to C'q(U) and let g : [0, go) x U —> R be a function in Cq(E) and let u : [0, go) x U —> R be a solution to the following problem: du 1 . r x „ < ~dt = 2 Au + 9 m [°> G0 ) x u ’ u is continuous on [0, go) x U and u( 0, x) = f(x). Moreover we assume that lim x -*b,xeu u (t, x) = 0 if b belongs to dU. Then u(t, x) = E x ( f(b(t )) : r > t) + E^ g(t — r, b(r))dr^J , where, as in Ex¬ ample 1 , r is the exit time of U. Also as in Example 1 , {b(s) : s 5 = 0} stands for //-dimensional Brownian motion. A proof can be given following the same lines as in the previous example. Example 3. (Feynman-Kac formula) Let U be an open subset of R", let / belong to C$(U) and let V : U —> R be an appropriate function and let u : [0, go) x U —*• R be a solution to the following problem: r ^ — = - Au — Vu in [ 0 , oo) x U: < dt 2 v i ) u is continuous on [0, go) x U and u(0,x) = f(x). Moreover we want that lim x ^b,xeu u (t, x) = 0 if b belongs to dU. Then u(t, x) = E x ^exp f 0 V (b(r))drj f(b(t)) : r > t). For the proof we fix t > 0 and we consider the process {M(s) : 0 ^ s < t} dehned by M (s) = u(t - s, b(s)) exp (- £ V(b(r))dr) l{ r>s } and we apply Ito’s formula to obtain: M(s) - M( 0) ri exp — r, b(r)) -\ —A u(t — r, b(r)) — V(b(r))u(t — r, b(r)) (J L Z + ^V(b(f,))dp S7u{t — r,b(r)) e Jo '-{r>r} dr JV(Kp))dp) l{r>r} • db(r) = J Vu(t - r, b(r)) exp J V(b(p))dpj l{ T>r } • db(r). Here we used the fact that u is supposed to be a solution of our initial value problem. It follows that the process {M(s) : 0 < s < t} is a martingale. Hence we may conclude that u(t,x) = E x [M(0)] = E x [. M(t )] / rt = E, = E t «( 0 , b(t)) exp J V(b(p))dpj : r > t f(b(t)) exp J V(b(p))dpj : t > t 181 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Example 4. (Cameron-Martin or Girsanov transformation). Let U be an open subset of R", let / belong to Cq(U) and let c \U —> EG be an appropriate vector field on U and let u : [0, oo) x U —*■ R be a solution to the following problem: = |A u + c.V« in [0, oo) x U] u is continuous on [ 0 , oo) x U and u(0,x) = f{x). Moreover we want that lim x ^b,xeu u {t, x) = 0 if b belongs to dU. Then u(t, x) = E x (exp (Z(t)) f(b(t)) : r > t), where Z(t) = f 0 c(b(r )) • db(r) - § \ c (K r ))\ 2 dr - For a proof we fix t > 0 and we consider the process {M(s) : 0 ^ s < t} defined by M(s) = u(t — s, b(s)) exp (Z(s)) 1{ T>S }. An application of Ito’s formula to the function f(s, x, y ) = u(t — s, x) exp (y) will yield the following result M(s) - M( 0 ) = J ~ r,b(r))exp(Z(r))l {T>r} dr rs + + + f Vu(t - r, b(r )) exp (Z(r)) l {r>r} • db(r) Jo I u(t — r, b(r)) exp (Z(r)) dZ(r ) Jo ■y pmin(s,r) f + 2 v r s 51 A u(t — r, b(r)) exp (Z(r)) 1 { T>r }dr Dju(t - r, b(r )) exp (Z(r)) 1 {r>r] d (bj, Z) (r) + 2 I'i ^ J u[t- r, b{r)) exp (Z(r)) 1 {T>r} d (Z, Z) (r) c)ijj 1 — ~^(t — r, b(r )) H—A u(t — r, b(r )) + c(b(r)) .V u(t — r, b(r )) (J L Z exp (Z(r)) l{ T>r }dr + + 2 f {Vu(t — r, b(r)) + u(t — r, b(r))c(b(r))} exp (Z(r)) l{ r>r } • db(r) Jo 7} J u(t- r, b(r )) exp (Z(r)) \c(b(r)) | 2 l {T>r} dr ^ J u(t — r, b(r)) exp (Z(r)) \c(b(r))\ 2 l {T>r} dr = f {S7u(t - r, b(r)) + u(t — r, 6 (r))c( 6 (r))} exp (Z(r)) l{ r>r } • db(r). Jo As above it will follow that u(t, x) = E^ (exp (Z(t)) f(b(t )) : r > t). Example 5. (Stochastic differential equation). Let (cr(x))J fc=1 , x e R", be a continuous square matrix valued function and let c(x) be a so-called drift vec¬ tor field (see the previous example). Suppose that the process (A x (s) : s ^ 0} 182 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales satisfies the following (stochastic) integral equation: X x (t)=x + f c(X x (s))ds+ f a(X x (s)) ■ db(s). Jo Jo In other words the process {X x (s) : s ^ 0} is a solution of the following stochas¬ tic differential equation: dX x (t) = c(X x (t))dt + a(X x (t)) ■ db(t ) together with X x (0) = x. The integral \ 0 a(X x ($))db($) has the interpretation f a(X x (s))db(s) = (2 f <?jk(X x {s))db k {s) Jo \ k=l Jo Next let u : [0,co) x l 1 ' ^ 1 be a twice continuously differentiable function. Then, by Ito’s lemma, u(t- s,X x (s)) -u(t,X x ( 0)) + f X7u(t-r,X x (r)) -dX x (r) Jo + \ 2 [ DfiMt-r,X*(r))i{X*,XI) (r). .<J-i J » Next we compute V d (.X x , XI) (r) = 2 a jm (. X x m(r )) a kn (. X x (r )) d (b m , b n ) (r) m,n— 1 = 2 <7 jm (X*(r))a km {X*(r))dr = (a(.X’(r))< r (X*{r)) T ) jh dr, m— 1 where a(x) T is the transposed matrix of a(x). Next we introduce the differential operator L as follows: -| IS V [Lf] (x) = - (■ a(x)a(x) T ) jk D j D k f(x ) + c j (x)D j f(x). j,k =1 J = 1 For our twice continuously differential function u we obtain: u(*-s,X*(s)) -u(t,X x (0)) = - J ^ (t-r,X x (r))dr + 2 fc,-(X*(r))D iU (t-r,X*(r))dr i=i + f Vm (f — r, X x (r)) g (X x (r)) ■ db(r) Jo 183 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales + \ 2 f (<t (X x (r)) a (X x (r)) T )j k D 0 D k u (t - r , X x (r)) dr Z j,k =i J o = J V'u (t — r, X x {r )) a ( X x {r )) • c?6(r) + J ^L — u(t — r, X x {r )) dr. So that, if yL — —J u = 0, then, for 0 ^ s < t, u(t-s,X x (s))-u(t,X x ( 0)) = T Xu(t-r,X x (r))a(X x (r))db(r), Jo and hence, the process M(s) := u(t — s, X x (s)) is a martingale on the interval [0, t]. It follows that u(t, x ) = E(M(0)) = E(M(t)) = E (u (0, X x (t ))) = E (/ pT(f))) where n(0,x) = f(x). For more details on stochastic differential equations see Chapter 4. OSRAM SYLVAN!A Light is OSRAM We do not reinvent the wheel we reinvent www.sylvania.com light. Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges. An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day. 184 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Example 6. (Quantum mechanical magnetic field). Let a be an appropriate vector field on W and let H (a, V) = \ (iV + a ) 2 + V be the (quantum mechan¬ ical) Hamiltonian of a particle under the influence of the scalar potential V in a magnetic Geld B{x) with vector potential a{x)\ i.e. B = V x a. Let / be a function in Co (KQ and let u : [0, go) x Mz/ —► M be a solution to the following problem: { r)i\ — = —H(a, V)u in [0, oo) x IT; u is continuous on [0, go) x R" and u(0,x) = f(x). Moreover we want that lim u(t,x ) = 0. Then u(t,x ) = [e z ^/(6(t))], where Z(t) = — if a(b(s )) • db(s) — -if V • a(b(s))ds — f V(b(s))ds Jo 2 Jo Jo V with V-d = V ——. Put Mis ) = uit — s, bis)) exp (Z(s)), 0 < s < t. An r-j ox* 3=1 J application of Ito’s formula to the function f(s,x,y ) = u(t — s,x) exp(y) will yield the following result M(s) - M( 0) = f(s, b(s ), Z{s)) - /(0, b( 0), Z( 0)) = J ^{a,b(a),Z(a))da + J V x f(a,b(a),Z(a))-db(a) + J ^(a,b(a),Z(p)) • dZ(a) + 1 J A x f(a,b(a),Z(a))da + \ t j, b W- ^ ^ M + | J o %), ^(o’))d (X (<t) = J (cr,6(cr),Z(cr))dcr + J V x f(a,b(a),Z(cr))-db(cr) + J /(cr,6(a),Z((7))-dZ(a)+1 J A x f(a,b(a),Z(a))da J Q fe(cr),^(cr))a i (6(<r)) da ^ j ° i=i = — — (t — a, 6(a)) e z ^da + (t — a, 6(a)) • d6(a) 185 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales + f u (t — cr, 6(cr)) e z ^ • dZ(a) + - f A x u (t — a, 6(a)) e z ^da Jo 2 J 0 v r s p — i ^ -r— (t — a,b(a)) e z ^aj{b{a)dc ]=x Jo v x j 1 + 2 rs v j u(t — a, b(a )) e V a,j (fr(cr)) 2 da Jo i=i ' rr i A JoH* +2^ u — ia(b(a)).W x u - \a(b(a))\ 2 u 2 1 —-zV • a{b{a))u — V(b(a))u [• (t — a, b(a)) e z ^da + f V x u (t — a, b(<j)) e z ^ ■ db(a) — i f u (t — a,b(a)) e z ^a(b(a)) ■ db(a) Jo Jo J j-y “ \ (*V + a) 2 - v| u (t - a, b(a)) e z(<T W + f \7 x u (t — a, 6(a)) e z ^ ■ db{a) — i f u (t — a,b(a)) e z ^a(b(a)) ■ db(a) Jo Jo = f \7 x u (t — a, b(a)) e z ^ ■ db(a) — i f u (t — a,b(cr)) e z ^a(b(a)) ■ db(cr). Jo Jo Here we used the fact that the function u satisfies the differential equation. The claim in the beginning of the example then follows as in Example 4. Example 7. A geometric Brownian motion (GBM) (occasionally called expo¬ nential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, also called a Wiener process: see e.g. Ross [116] Section 10 . 3 . 2 . It is applicable to mathematical modelling of some phenomena in financial markets. It is used particularly in the field of option pricing because a quantity that follows a GBM may take any positive value, and only the fractional changes of the random vari¬ ate are significant. This is a reasonable approximation of stock price dynamics except for rare events. A stochastic process S t is said to follow a GBM if it satisfies the following stochastic differential equation: dS(t) = fiS{t) dt + aS(t) dW(t) where W(t) is a Wiener process or Brownian motion and // (“the percentage drift” or “drift rate”) and a (“the (percentage or ratio) volatility”) are constants. For an arbitrary initial value S(0) the equation has the analytic solution S(t) = S (0) exp ((/X - y) t + crW(t)) , which is a log-normally distributed random variable with expected value given by E[<S(t)] = e tlt S{ 0) and variance by Var (S(t)) = e 2 ^S{ 0) 2 (e^ — 1 ^. 186 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales The correctness of the solution can be verified using Ito’s lemma. The random variable log is normally distributed with mean (// — |cx 2 ) t and variance <r 2 f, which reflects the fact that increments of a GBM are normal relative to the current price, which is why the process has the name “geometric”. Example 8. The term Black-Scholes refers to three closely related concepts: 1. The Black-Scholes model is a mathematical model of the market for an equity, in which the equity’s price is a stochastic process. 2. The Black-Scholes PDE is a partial differential equation which (in the model) must be satisfied by the price of a derivative on the equity. 3. The Black-Scholes formula is the result obtained by solving the Black- Scholes PDE for a European call option. Fischer Black and Myron Scholes first articulated the Black-Scholes formula in their 1973 paper, “The Pricing of Options and Corporate Liabilities.”: see [19]- The foundation for their research relied on work developed by scholars such as Jack L. Treynor, Paul Samuelson, A. Janies Boness, Sheen T. Kassouf, and Edward O. Thorp. The fundamental insight of Black-Scholes is that the option is implicitly priced if the stock is traded. Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term “Black-Scholes” options pricing model. Merton and Scholes received the 1997 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for this and related work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy. 187 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 7. Black-Scholes model The text in this section is taken from Wikipedia (English version). The Black- Scholes model of the market for a particular equity makes the following explicit assumptions: 1. It is possible to borrow and lend cash at a known constant risk-free interest rate. 2. The price follows a geometric Brownian motion with constant drift and volatility. 3. There are no transaction costs. 4. The stock does not pay a dividend (see below for extensions to handle dividend payments). 5. All securities are perfectly divisible (i.e. it is possible to buy any frac¬ tion of a share). 6. There are no restrictions on short selling. 7. There is no arbitrage opportunity. From these ideal conditions in the market for an equity (and for an option on the equity), the authors show that it is possible to create a hedged position, consisting of a long position in the stock and a short position in [calls on the same stock], whose value will not depend on the price of the stock. Notation. We define the following quantities: - S, the price of the stock (please note as below). - V(S,t), the price of a financial derivative as a function of time and stock price. - C(S, t ) the price of a European call and P(S, t ) the price of a European put option. - K, the strike of the option. - r, the annualized risk-free interest rate, continuously compounded. - fj,, the drift rate of S, annualized. - <7, the volatility of the stock; this is the square root of the quadratic variation of the stock’s log price process. - t a time in years; we generally use now = 0, expiry = T. - II, the value of a portfolio. - R , the accumulated profit or loss following a delta-hedging trading strategy. - N(x) denotes the standard normal cumulative distribution function, 1 12 - N'(x) = , _ e~* x denotes the standard normal probability density V 2n function. 188 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Black-Scholes PDE. Simulated Geometric Brownian Motions with Parameters from Market Data In the model as described above, we assume that the underlying asset (typically the stock) follows a geometric Brownian motion. That is, dS(t) = /nS'(f) dt + crS(t) dW(t ), where W(t) is a Brownian motion; the dW term here stands in for any and all sources of uncertainty in the price history of a stock. The payoff of an option V(S,T ) at maturity is known. To find its value at an earlier time we need to know how V evolves as a function of S and T. By Ito’s lemma for two variables we have m) , t) - *y) ds(t) + ® * + l JdimA d(s, S) (t) CD Cl Z C Z D + AlflA + ^ gp '| * as dt d£ 2 .dV(S(t),t) „ jr , , 3 rx5(/)- ; dlT(f). (3.208) Now consider a trading strategy under which one holds a(f) units of a single option with value iS'(t) and b(t) units of a bond with value ft(t) at time f. The value V ( S(t),t ) of the portfolio of the trading strategy (a(t), 6(t)) is then given by V (. S(t),t ) = a(t)S(t ) + b(t)(3(t). (3.209) Observe that (3.209) is equivalent to V ( S(t),t ) — a(t)S(t ) 6(f) = /3(f) In addition, a(t) = ^ which i s called the delta hedging rule. Assum- ds ing, like in the Black-Sholes model, that the strategy (a(t), 6(f)) is self-financing, which by definition implies dV () = a(f) dS'(t) + 6(t) d/3(t), (3.210) we get dV(t) = dt + b(t) d/3(t) + cra(t)S(t) dW(t). (3.211) Assume that the process f > /3(t), he., the bond price, is of bounded varia¬ tion. By equating the terms with dW(t) in (3.208) and (3.211) we see a(f) = —— ~ ~ Ftoni this and again equating the other terms in (3.208) and (3.211) and using (3.209) we also obtain ( « + 1 M - b(t ) dm - (v ( S(t ), t) - g(t) Al . (3,212) 189 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales If the interest rate for the bond is constant, z.e., if d/3(t) = rj3(t)dt, or, what amounts to the same, (3(t) = /3(0)e rt , then from (3.212) it also follows that SV(S(t),t) 1, 2 d 2 V(S(t),t) - + ;&W - dt = rlv(S(t),t)-S(t) dS 2 dV (S(t),t) ds (3.213) If we trade in a single option continuously trades in the stock in order to hold 8V — shares, then at time t, the value of these holdings will be U(t) = V(S(t),t)-S(t) 8V ( S(t),t ) Is The composition of this portfolio, called the delta-hedge portfolio, will vary from time-step to time-step. Let R(t) denote the accumulated profit or loss from following this strategy. Then over the time period [t,t + dt], the instantaneous profit or loss is dR(t) = dV ( S(t),t ) dV(S(t),t) 8S dS(t). By substituting in the equations above we get dR(t) = dV(S(t),t) , 1 2c , 2 d 2 V(S(t),t) 0t + * as 2 dt. This equation contains no dW[t) term. That is, it is entirely risk free (delta neutral). Black, Scholes and Merton reason that under their ideal conditions, the rate of return on this portfolio must be equal at all times to the rate of return on any other risk free instrument; otherwise, there would be opportunities for arbitrage. Now assuming the risk free rate of return is r we must have over the time period [t,t + dt] (Black-Scholes assumption): rll(t) dt = dR(t) dv(s(t),t) , i 2c2 en'(S(t),t) dt + 2° b dS 2 dt. Observe that the Black-Sholes assumption comes down to the assumption of self-financing, because the results If we now substitute in for II(£) and divide through by dt we obtain the Black-Scholes PDE: 8V (S(t),t) dt i 1 2 q : + -a!S 2 d 2 V(S(t),t) +rg dV(S(t),t) 8S 2 dS rV(S(t),t) = 0 . (3.214) Observe that the Black-Sholes assumption comes down to the assumption of self-financing, because the resulting partial differential equation in (3.213) and (3.214) is the same. With the assumptions of the Black-Scholes model, this partial differential equation holds whenever V is twice differentiable with respect to S and once with respect to t. Above we used the method of arbitrage-free pricing (“delta-hedging”) to derive some PDE governing option prices given the Black-Scholes model. It is also possible to use a risk-neutrality argument. This latter method gives the price as the expectation of the option payoff under a particular probability measure, called the risk-neutral measure, which differs from the real world measure. 190 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Black-Scholes formula. The Black-Scholes formula is used for obtaining the price of European put and call options. It is obtained by solving the Black-Scholes PDE as discussed - see derivation below. The value of a call option in terms of the Black-Scholes parameters is given by: C{S , t) = C ( S(t),t ) = S(t)N(<h) - Ke- r{T ~ t) N(d 2 ) with (3.215) log® + (r + t) ( T ~ *) ,_ d\ = - v , - - and d 2 = d\ — aVT — t. (3.216) o\/T — t The price of a put option is: P(S, t) = P ( S(t),t ) = Ke~ r{T -^N{-d 2 ) - S(t)N{-d{). (3.217) For both, as above: 1. N(-) is the standard normal or cumulative distribution function. 2. T — t is the time to maturity. 3. S = S(t) is the spot price of the underlying asset at time t. 4. K is the strike price. 5. r is the risk free interest rate (annual rate, expressed in terms of con¬ tinuous compounding). 6. a is the volatility in the log-returns of the underlying asset. Deloitte Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Interpretation. The quantities N ( d \) and N (d 2 ) are the probabilities of the option expiring in-the-money under the equivalent exponential martingale prob¬ ability measure (numeraire = stock) and the equivalent martingale probability measure (numeraire = risk free asset), respectively. The equivalent martingale probability measure is also called the risk-neutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure. Derivation. We now show how to get from the general Black-Scholes PDE to a specific valuation for an option. Consider as an example the Black-Scholes price of a call option, for which the PDE above has boundary conditions (7(0, t) = 0 for all t C(S, t) -> S as 5 — oo C(S, T) = max(S' — K,0). The last condition gives the value of the option at the time that the option matures. The solution of the PDE gives the value of the option at any earlier time, E [max(S' — K, 0)]. In order to solve the PDE we transform the equation into a diffusion equation which may be solved using standard methods. To this end we introduce the change-of-variable transformation o 2 t = T — t, u(x,t ) = C (Ke x -( r ~ ) r ,T — r) e rT , and x = log — + (r-)r. V / K 2 Note: in fact in case we consider a call option we replace V ( S(t),t ) with C (S(t),t). Instead of u we may also consider v(x,t) = V (Ke x -( r -* a2 ) iT ~ t) ,t) e r(T - f) . In case we consider a European call option we take as final value for v: v(x, T) = V(Ke x ,T) = C(Ke x ,T ) = max (Ke x — K, 0) = K max (e* — 1, 0). Then the Black-Scholes PDE becomes a diffusion equation du 1 2 d 2 u dr 2 a dx 2 The terminal condition C(S,T ) = max(,$' — K, 0) now becomes an initial con¬ dition u(x, 0) = uo(x) = K max ( e x — 1,0). Using the standard method for solving a diffusion equation we have u 0 (y)e- ix - y)2/i2a2r) dy. After some calculations we obtain u(x, T ) = a V2 7 TT J- 1 u(x,t) = Ke x+a2T/2 N (di) - KN (d 2 ) where . x + o 2 t X d\ = -—— and (I 2 ~ — 7 =. cryr cr^/r 192 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Substituting for u, x, and r, we obtain the value of a call option in terms of the Black-Scholes parameters is given by C(S,t) = SN(di) - Ke~ r{T - t) N(d 2 ), where d\ and d 2 are as in (3.216). The price of a put option may be computed from this by the put-call parity and simplifies to P(S,t) = Ke~ r{:r - t) N(-d 2 ) - SN(-di). Risk neutral measure. Suppose our economy consists of 2 assets, a stock and a risk-free bond, and that we use the Black-Scholes model. In the model the evolution of the stock price can be described by Geometric Brownian Motion: dS ( t ) = /rS ( t ) dt + crS ( t ) dW ( t ) where W ( t ) is a standard Brownian motion with respect to the physical measure. If we define W(t) = W(t ) + Girsanov’s theorem states that there exists a measure Q under which W(t) is a standard Brownian motion, i.e., a Brownian motion without a drift term and such that Eq theorem, whic 4.24 in Chapter 4 Section 3. The quantity u is m W(t) 2 = t. For a more thorough discussion on the Girsanov’s act (much) more general, see assertion (4) in Proposition is known as the market price H — r of risk. Differentiating and rearranging yields: dW(t) = dW(t ) - -—- dt. G Put this back in the original equation: dS(t) = rS(t) dt + <rS(t) dW(t). The probability Q is the unique risk-neutral measure for the model. The (dis¬ counted) payoff process of a derivative on the stock H(t) = Eq (H(T) | T t ) is a martingale under Q. Since S and H are Q-inartingales we can invoke the martingale representation theorem to find a replicating strategy - a holding of stocks and bonds that pays off H(t) at all times t < T. The measure Q is given by <5(^4) = E [e -z ( T )l^], A e where y-r G w(t). In fact a more general result is true. Let s r'T that E exp \h(s)\ 2 ds < GO. Put h(s) be a predictable process such Zh(t) = f h(s)dW(s) + ^ f \h(s)\ 2 ds. Jo 2 Jo Define the measure Qh by Qh(A) = E[e -Zh ^l^], A e Put 114(t) = W(t) + h(s) ds. Then the process 114 is a Brownian motion relative to the measure Qh- The proof of this result uses Levy’s characterization of Brownian 193 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales motion: see Corollary 4.7. It says that a process Wh is a Q/,-Brownian motion if and only if the following two conditions are satisfied: (1) The quadratic variation of Wh satisfies (Wh, Wh) (t) = t. (2) The process Wh is a local martingale relative to the measure Q/- ( . (For a proof of this result see Theorem 4.5.) In our case we have (Wh, Wh) (t) = {W, W) ( t ) = t, and so (1) is satisfied. In order to establish (2) we use Ito calculus to obtain: e~ Zh ®W h (t) f e~ Zh{s) dW(s) - f Jo Jo Zh{s) W h (s)h(s)dW(s). Since the process t ^ e Zh(t> is a martingale we see that the process IF/, is a local Q/r m artingale. We like to spend more time on the Black-Sholes model and the corresponding risk-neutral measure. Again we have trading strategy ( a(t),b(t )) of a financial asset and a bond. Its portfolio value V(t) := V ( S(t),t ) is given by V(t) = a(t)S(t ) + b(t)/3(t). Here S(t) is the price of the option at time t and (3(t) is the price of the bond at time t. It is assumed that the process t > S(t) follows a geometric Brownian motion: dS(t) = nS(t) dt+aS(t) dW(t), or S(t) = S( Let S(t) be the discounted price of the option, i.e., m m f3(t) S(t). (3.218) Put W(t) = W(t) + q(s ) ds, where q(s) Then the process S(t) satisfies the equation dS(t) = cr ^}^} S(t)d (— f (fi — ^ ds + IF^)^ = <rS(t) dW(t). (3.219) PW V^Jo V p(s )) ) Put Z q (t ) = I q(s)dW(s) + - j q(s) 2 ds. By Girsanov’s theorem the process ^ J° 2 Jo t i—> W(t) is a (standard) Brownian motion under the measure Q q given by Q q (A) = E [e _z 9( T )i A ] ; A e Tt- The solution S(t) of the SDE in (3.219) can be written in the form S(t ) = S( 0)e aW(t) -^ H . Assume that the portfolio is self-financing we will show that (3(t) V(t) = m h(S(T))\5 t t e [0, T], (3.220) (3.221) where V(T) is equal to the contingent claim h (S(T)) at the time of maturity T. Of course, E® 9 [F | T t ] denotes the conditional expectation of F relative Q q , given the a-field = a (W(s) : s ^ t) of the variable F e L 1 (0, 5Fr,Qg) with respect to the probability measure Q q . Another application of Ito’s lemma 194 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales together with the definition of S(t), W(t) and V(t) = ^ ^-V(t) shows the following result dv(t)= mm Pit) V(t) dt + dV(t) Pit) 2 /3(t) (the hedging strategy ( a(t),b(t )) is self-financing) = ~ v ( f ) dt + dS(t) + b(t)d(3(t)) (employ the equation for the option price Sit)) )P ( ) y(f^ dt _|_ V 1 s(t) fiia(t ) dt + b(t)^-y -y dt + adW ( t ) m 2 Pi® )P'(t) Pit) 2 Pi 0 ) + Pi*) ( a(t)S(t ) + b(t)/3(t )) dt m pit) pp) MO) S(t) ( jia{t) dt + b{t ) t— 7 dt + adW(t) aa(t) t- : 5'(t) d -j — Pit) = a ait) S(t) dWit). Pit) P'i 3 ) Pis) ds + W(t) (3.222) SIMPLY CLEVER SKODA We will turn your CV into an opportunity of a lifetime Do you like cars? Would you like to be a part of a successful brand? We will appreciate and reward both your enthusiasm and talent. Send us your CV. You will be surprised where it can take you. 195 Send us your CV on www.employerforlife.com Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In fact the equality in (3.222) could also have been obtained by observing that d,V(t) = a(t) dS(t), and dS(t) = adW(t). (3.223) From (3.222) we infer V(t) = V (0) + <7 f a(s)dW(s), Jo (3.224) and hence, the process t <-*■ V(t) is a martingale with respect to the measure Q , q . So from (3.224) we get m- p(t) and hence V(t) = V(t) = Pit) V(T) % = £^9 m P(T) V{T) % V(t) = E Q « V(T) I % vm In addition, we observe that P(T) e -±* 2 (T-t)+a(W(T)-W(t)) Pit) im h(S(T))\% (3.225) (3.226) = S (t) exp f( P\s) 1 2 p{s) 2 a ds + a ( W(T ) - W{t )) S(0) exp ( crW(t ) + ( /i -cr 2 1 t exp f( 1 _ - 2 Pis) 2 ds + CT (w(T) - W(t) + | q(s)ds = S( 0) e ( ti - 1 ^ 2 ) T+aW{T) = S(T). Inserting the equality for S(T) from (3.227) into (3.226) yields V(t) = Pit) PiT)‘ h f S(t) e -^ 2 (T-t)+a(w(T)-W(t)) (3.227) (3.228) Since the variable S(t) is measurable with respect to %, Since process t > W(t) is a Q^-Brownian motion, the variable W(T ) — W(t) and the cr-field 5) are Q q - independent. Moreover, the process t >—> (3(t) is supposed to deterministic. Hence, since the variable S(t) is measurable with respect to we deduce that V(t) = V(S(t),t) 1 , — . — ' ■ (3.229) a/ 27f J— r &L h ( x Pi^l e -^(T-t)+a^T=iy\ -\y* > J-oc p(t) ^ m ) y I x=S(t) Hence if the pay-off, i.e. the value of the call option at expiry (time of maturity T), is given by h(S(T )) = max {S(T) — K, 0}, then the value of the portfolio at time t < T is given by the formula in (3.229). If (3(t) = f3(0)e rt , then this integral can be rewritten as in (3.215) with C (S, t) = C (. S(t),t ) = V ( S(t),t ) = V(t). Similarly, if h ( S(T )) = max {K - S(T), 0}, then P (S, t) = P ( S(t),t) = V ( S(t),t ) = V(t) is the price of a European put option: see the somewhat 196 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales more explicit expression in (3.217). For a modern treatment of several stock price models see, e.g., Gulisashvili [ 60 ]. 8. An Ornstein-Uhlenbeck process in higher dimensions Part of this text is taken from [ 146 ]. Let C (t, s), t ^ s, t, s e E, be a family of d x d matrices with real entries, with the following properties: (a) C(t, t) = I, t e E, (/ stands for the identity matrix). (b) The following identity holds: C(t, s)C(s, r) = C(t. t) holds for all real numbers t, s, r for which t ^ s ^ r. (c) The matrix valued function (t, s, x ) > C(t , s)x is continuous as a func¬ tion from the set {(fsjeiGlh t ^ s} x R d to R d . Define the backward propagator Yc on Cb (R d ) by Yc(s,t)f(x ) = f (C(t, s)x), x e E d , s < t, and / e Ci„ (E d ). Then Yc is a backward propagator on the space Cb (R d ), which is a (Cb (E d ) , M (E d ))-continuous. Here the symbol M (E d ) stands for the vector space of all signed measures on E'k The operator family {Fc(s, t) : s < t} satisfies Y c (si, s 2 ) Y c (s 2 , s 3 ) = Y c (si, s 3 ), Si < s 2 < s 3 . Let W(t) be standard m-dimensional Brownian motion on (f2,T t ,P) and let a(p) be a deterministic continuous function which takes its values in the space of d x m-matrices. Put Q(f>) = a(p)a(p)*. Another interesting example is the following: Yc,q ( s,t ) f(x) = ^d /2 J e ~^ lv]2 f s)a; + Q C(t,p)Q(p)C(t,p)*dp S j = E / C(t,s)x + f C(t,p)a(p)dW(p) J S dy (3.230) where Q(p) = a(p)a(p)* is a positive-definite d x d matrix. Then the propaga¬ tors Yc,q and Yq,s are backward propagators on Cb (E d ). We will prove this. The equality of the expressions in (3.230) is a consequence of the following ar¬ guments. Let the variable £ 6 E d have the standard normal distribution. Fix t Y t. Both variables t ^ t, and t Y r, (3.231) X T,x (t) := C (t,r) x + J C (: t , p) <r(p)dW(p), C(t, t)x + ( f C(t, p)Q(p)C(t, p)*dp\ £, are E d -valued Gaussian vectors. A calculation shows that they have the same expectation and the same covariance matrix with entries given by (3.242) below with s = t. Next suppose that the forward propagator C on R d consists of contractive op¬ erators, i.e. C(t, s)C(t, s)* ^ I (this inequality is to be taken in matrix sense). 197 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Choose a family S(t,s) of square d x d-matrices such that C(t, s)C(t, s)* + S (£, s) S (t, s)* = /, and put Y c ,s(s,t)f(x ) |y|2 / (C(t, s)a; + Sft, s)y) dy. (3.232) In fact the example in (3.232) is a special case of the example in (3.230) provided Q(p) is given by the following limit: Q(p) = lim HO I-C(p-h)C(p-hY h (3.233) If Q(p) is as in (3.233), then S (t, s ) S (■t , s)* = I -C (t, s) C (■ t , s)* = f C (t, p) Q(p)C (t, p)* dp. J s The following auxiliary lemma will be useful. Condition (3.234) is satisfied if the three pairs (C x , S x ), {C 2 , S 2 ), and (C 3 , S 3 ) satisfy: C^+S^* = C 2 C*+S 2 S* = CaC'l + S 3 S% = I. It also holds if C 2 = C {t 2 ,ti), and S 3 S* = f C (tj,p)a(p)a(p)*C (tj,p)* dp, j = 1,2, and Jtj -1 *5.3*S'! = f C (t 2 ,p)a(p)a(p)*C (t 2 ,p)* dp. MAERSK I joined MITAS because I wanted real responsibility The Graduate Programme for Engineers and Geoscientists www.discovermitas.com Real work International opportunities Three work placements a I was a construction supervisor in the North Sea advising and helping foremen solve problems 198 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.89. Lemma. Let C\, S x , C 2 , S 2 , and C 3 , S 3 be dxd-matrices with the following properties: C3 = c 2 c u and q c<* , q n* n rr* 02010^02 ~r 02^2 — O3O3 . (3.234) Let feC b ( R d ) , and put Y lt2 f(x) = 1 (27T) d/2 J e“^ |y|2 / (Cl® + 3 iy) dy; (3.235) Y 2 , 3 f(x) = 1 (27T) d/2 J e _ ^ |y|2 / (C 2 x + S 2 y)dy; (3.236) Yi,sf(x) = 1 ( 2 yr ) d/2 J e~^ lyl2 f (C 3 x + S 3 y)dy. (3.237) Then W,2^2,3 = Y lt 3. PROOF. Let the matrices Cj and Sj, 1 C j < 3, be as in (3.234). Let f g C b (R d ). First we assume that the matrices S x and Co are invertible, and we put A 3 = S’7 1 Cf 1 S3 , and A 2 = Sf 1 Cf 1 So . Then, using the equalities in (3.234) we see A 3 A 3 = I + A 2 A;i;. We choose a d x d-matrix A such that A* A = I+A 2 A 2 , and we put D = (A~ 1 )* A 2 A 3 . Then we have A 3 A 3 = I+D*D. Let / e C b (M d ). Let the vectors (yi,y 2 ) e x and (y, z)eR J x be such that (3.238) Since A 2 A 2 (/ + A 2 A 2 ) = A 2 (1 + A 2 A 2 ) A|, we obtain det (/ + A 2 A*) = det (/ + A*A 2 ). Hence, the absolute value of the determinant of the matrix in the right-hand side of (3.238) can be rewritten as: det A 3 —A 2 A 0 A - 1 -1 = | det A 3 (det A) -11 2 det (A 3 A 3 ) det (/ + A 2 A£) det (A*A) det (d -I- A^A 2 ) = 1 . (3.239) From (3.238) and (3.239) it follows that the corresponding volume elements satisfy: dtj\ dy 2 = dydz. We also have |di | 2 + 1 2 / 2 1 2 = \y \ 2 + \z~ Dy \ 2 . (3.240) Employing the substitution (3.238) together with the equalities diji dy 2 = dy dz and (3.240) and applying Fubini’s theorem we obtain: Yi, 2 Y 2 y 3 f(x) = JJ e - K | w i| 2 + l» 2 | 2 ) j (C 2 C 1 X + C 2 Siy x + S 2 y 2 ) dy x dy 2 = (2JJ e_ K |2/|2+|z " jD2/|2 )/((F 3 x + S 3 y ) dydz 199 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales = nfjw J + d V = (3-241) for all / 6 Cb (M d ). If the matrices Si and C-i are not invertible, then we replace the C\ with C\ )£ = e~ £ C\ and ,5'i ;£ satisfying Ci t£ C* f + Si j£ S* e = /, and lim e |o<S'i, e = -S'i- We take 62 ,e = instead of S 2 . In addition, we choose the matrices C 2 , e , s > 0, in such a way that £ + £ = /, and hm £ |o C2,e = C *2 • This completes the proof of Lemma 3.89. □ We formulate a proposition in which an Ornstein-Uhlenbeck process plays a central role. Here p •—* a(p) is a deterministic square matrix function, and < 2 (p) = 3.90. Proposition. Put X T,x (t ) = C (t, t)x + C (t, p) a(p)dW(p) . Then the process X T,x (t) is Gaussian. Its expectation is given by E[X T,a: (t)] = C (t,r)x, and its covariance matrix has entries (s, t 5 = r J C( S ,p)Q(p)C(t,p)*dpJ (3.242) Let {(O, 9\ P T;X ), (X(t), t ^ 0 ), (M d , 23 d )} 6 e the corresponding time-inhomogen- eous Markov process. By definition, the P -distribution of the process t « X T,x (t), tp r, is the P T)a: - distribution of the process 1 1 —> X(t), t ^ r. Then this process is generated by the family operators L(t), t 5= 0, where d mm - \ 2 QiAt)DjD k f(x) + (Vf(x),A(t)x). z j,k =1 Here the matrix-valued function A(t ) is given by A(t ) = hm The semigroup e sL<yt \ s ^ 0, is given by (3.243) C(t + h,t) — I h ’ e sL{t) f(x) = E / + J e (s - p) ^ (t) (j(t)dW(p) ^ J | e ^) I+ (JeM( t )g (t ) e MWg p j ' y \ dy (2tt) = f P (^, n?, ?/; t) /(y)dy (3.244) where, with QA(t )( s ) = e pA ^Q(t)e pA ^* dp, the integral kernel p(s,x,y,t) is Jo given by P(s,x,y,t ) ( 2 vr ) d/2 (detg A(t) (s)) d/2 - § ( (QA(t) (s)) 1 ( y-e sA ^x ) ,y-e aA toxy) 200 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales If all eigenvalues of the matrix A(t ) have strictly negative real part, then the measure B ^ \jdJ 2 Jc _ ^ l ^ /|2 1b (J e pA{t p{t)e pA{t) *dpfj dy defines an invariant measure for the semigroup e sL ^\ s ^ 0. A Markov process of the form {(f2, P T>a; ), (X(t),t ^ 0), (M d ,!B R d)} is called a (generalized) Ornstein-Uhlenbeck process. It is time-homogeneous by putting C(t, s ) = e ~d-s)A^ w j iere ^4 is a square d x d-matrix. We will elaborate on the time-homogeneous case. In this case we write, for x, b e M d . r-t S{t)f{x) := E / e~ tA x + (I- e~ tA )6+ r Jo 0 -(t-s)A a dB(s) (3.245) where / : M. d —> C is a bounded Borel measurable function. If / belongs to Co (E d ), then S(t)f does so as well. For brevity we write X x (t) = e ~ tA x + (I - e~ tA )» + f Jo 0 -(t-s)A adW(s). Because achieving your dreams is your greatest challenge. IE Business School’s Master in Management taught in English, Spanish or bilingually, trains young high performance professionals at the beginning of their career through an innovative and stimulating program that will help them reach their full potential. Choose your area of specialization. Customize your master through the different options offered. Global Immersion Weeks in locations such as London, Silicon Valley or Shanghai. Because you change , we change with you . www.ie.edu/master-management mim.admissions@ie.edu f # In YwTube ii Master in Management • Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales It also follows that for such functions lim^o S(t)f(x) = f(x) for all x e R d . Since we also have the semigroup property S (ti + t 2 ) f = S (ti) S (t 2 ) f for all ti, t 2 ^ 0 , it follows that the semigroup t >—► S(t) is in fact a Feller semigroup. Theorem 3.37 implies that there exists a time-homogeneous Markov process {(fi,T,P x ) xeRd ,(X(t),t> 0), (d t ,t ^ 0), (R d , B R d)} such that for a bounded Borel function / we have Ex [/ (*(*))] = E [/ (X x m = S(t)f(x), X e R d . (3.246) Nest we prove the semigroup property. First we observe that, for x e M. d and *i, t 2 > 0, X s (ti + t 2 ) = e~ t2A X x (ti) + (I - e~ t2A ) b+ f * e~^~ s)A a dW (s + h ). (3.247) Jo Let (Vt w be the probability space on which the process t >—► W(t) is a Brownian motion. Let (ff) l ) />0 be the internal history of the Brownian motion {W(t) : t ^ 0 }, so that = a (VF(s) : s < t). Then by the equality in (3.247) we have E [/ (X* (h + t 2 )) | 3%] (3.248) = E f (^e~ t2A X x (h) + (I- e ~ t2A ) b + J 2 e-( t2 ~ s)A adW (s + h) rrW We employ the fact that the state variable X x ( t \) is J^-measurable, and that rti+t 2 l e -(ti+t 2 -s)A (J dW(s ) = f 2 e- (f2 - s)A ad{W{s^t 1 )-W{t 1 )} Jo is P-independent of T))' 7 and possesses the same P-distribution as the variable J* 2 e~^ t2 ~ s ^ A a dW (s) to conclude from (3.248) the following equality: E [/ (X* (h + t 2 )) | T* 7 ] = E f (e~ t2A z + (I- e ~ t2A ) b + J 2 e~ {t2 ~ s)A adW (s) \z=X x (ti) = K[f(X‘ (i 2 ))] |„ x , (ti) • (3,249) From (3.249) it follows that the process t <—>■ X x (t) is a Markov process and that, by the definition of the operators S(t), t 5= 0, S(t 1 + t 2 )f(x)=E[f(X x fa +t 2 ))] E E [f (X z (t 2 ))] I Z=X*(t!) E [S(t 2 )f(X x fa))] = S (ti) S (t 2 ) f(x). (3.250) We calculate the differential dX x (t ) and the covariation process (X ^, X x 2 y (t): dX x (t) = —A(X x (t) — b) dt + a dW(s), and (3.251) {X x v X x 2 ){t) = \ (e- sA aa*e- sA *) ds = cov (X? (*),X?(f)) . (3.252) Jo V ' 31,32 202 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In other words the process t >—* X x (t) satisfies the equation X x (t)=x + f A(b-X x (s )) ds+ f odW(t Jo Jo (3.253) Since its covariation is deterministic we have that the covariation coincides with its covariance: see (3.252). Let / : R. d —> C be a bounded continuous function with bounded and continuous first and second order derivatives. Next we apply Ito’s lemma, and employ (3.251) and (3.252) to obtain f (X x (t)) - f (. X*(0 )) = f V/ (X x (s)) • {-A (X x (s) -b)}ds J 0 d rt + 2 + ) t, f D h D h f(X’( S ))(e- A aa’e- A ’) d S 2 h,h- i Jo n ' n f X7f(X x (s)) -adW(s). (3.254) Jo Upon taking expectations in the right-hand and left-hand sides of (3.254), using the fact that the stochastic integral in (3.254) ia a martingale, and letting t [ 0 shows: L A m := Um S -AM - m _ lim 1 J v ’ no t no t -< d = -(A(x-b))-Vf(x) + - 2 (3.255) 31,32=1 In the following proposition we collect the main properties of the time-homo¬ geneous Ornstein-Uhlenbeck process t >—> X x (t). It is adapted from Proposition 3.90. In adition, a = a(p) is independent of p. 3.91. Proposition. Put X x (t) = e~ tA x + (/ - e~ tA ) b + f 0 e~ (t ~ p)A adW(p). Then the process X x (t ) is Gaussian. Its expectation is given by E[dP(t)] = e~ tA x + (/ — e~ tA ) b, and its covariance matrix has entries .in (s,t) \ e -(s- P )A a(T *e-(t- P ) A * df) (3.25 6 ) ' 31,32 Let {(U, lb, P T;X ), (X(t), t ^ 0), (M d , 23 d )} be the corresponding time-inhomogen- eous Markov process. By definition, the P -distribution of the process t > X x {t), t ^ t, is the P x -distribution of the process t > X(t), t ^ 0. Then this process is generated by the operator La, t ^ 0, where 1 d LaJ(x) = - J] (ar) jiJi D JI D J J(x)-{Vf(x),A(x-b)). (3.257) 31,32 = 1 / r P-cov(A-( s ),XJ(t)) = M The semigroup e sLA , s ^ 0, is given by 0 sL a f(x) = E f (^e- sA (x - b) + b + J e-( s ~ p)A a dW{p) 203 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales = ^ ^ J e ^ ly|2 f aA ( x ~b) + b+ e pA ov*e pA * dp^j yj dy = J p A (s,x,y) f{y)dy (3.258) where, with Qa(s ) = e~ pA aa*e~ pA * dp, the integral kernel pa ( s,x,y ) is given Jo by Pa ( s,x,y ) 3 ( - !(( < 2'4( s )) 1 (y~ e sA (x-b)-b),y-e sA (x-b)-b )) (2Tv) d/2 (det Q A (s)) d/2 If all eigenvalues of the matrix A have strictly positive real part, then the measure B h- > -— ^yi /2 j e~^ 1 b (^> + J" e~ pA aa*e~ pA * y dp'j dy (3.259) defines an invariant measure for the semigroup e sLA , s ^ 0. Proof. The results in Proposition 3.91 follow more or less directly from those in Proposition 3.90. The result in (3.259) follows by letting s — > go in the second equality of (3.258) or in the definition of the probability density Pa (s, x, y). □ For more information about invariant, or stationary, measures see, e.g., [146] (Chapter 10) and the references therein like Meyn and Tweedie [97]. In order to apply our results on the Ornstein-Uhlenbeck process to bond pricing and determining interest rates in financial mathematics the identities and results in the following proposition are very useful. It will be applied in the context of the Vasicek model. 3.92. Proposition. Let the notation and hypotheses be as in Proposition 3.91. Put Aft, T) = J e~ pA dp = | e~ ps ds = A -1 (/ - e ~ {T - t)A ), 0 <t^T, where the last equality is only valid if A is invertible. Let y be a vector in M. d . The following assertions hold true. (1) The following identity is true for 0 < t < T: E J X x {s)ds = A(t,T){X x (t)-b) + {T-t)b + J A (p, T) a dW(p). (3.260) (2) The random vector X x (s) ds is Gaussian (or, what is the same, mul¬ tivariate normally distributed) with conditional expectation given by J X x (s)ds\3 r t =E J X x (s)ds\X x (t) = Aft, T) (X x (t) — b) + (T—t)b, (3.261) 204 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales and covariance matrix given by (1 ^ ji, j 2 ^ d) cov | X^s)ds\X x (t),j t X x 2 (s)ds\X x (t ) f A (p, T) aa*A (p, T)* dp\ . (3.262) Jt / n i do (3) TTie random variable (y , X x (s) dsj is normally distributed with con¬ ditional expectation given by r T E (y, X x (s)dsj\? t = E (y,^ X x (s)dsJ\X x (t) = (y,A(t,T) (X x (t)-b)) + (T-t) (y, b ), and variance given by (3.263) V, \t I = \<r*A(p,T)*yf dp. (3.264) (4) The conditional expectation of exp (y, ^ X x (s) ds ^ given is log¬ normal, and E exp - < y /,j> s) ds ) ) 35 = E exp y,| T Xk5)ds^ | X x (t) exp <y, A(i, T) (X x (t) - b )) — (T — t) (y, b)+ l - £ \<j*A (p, T)* yf dp) . (3.265) 205 Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Proof. (1) From (3.247) we see, for s ^ t, A x (s) = e~ {s ~ t)A (X x (t) - b) + b + | e~ {s - p)A a dW (p ). (3.266) Then we integrate the expressions in (3.266) against s for t < s < T, and we interchange the integrals with respect to ds and dW(p) to obtain the equality in (3.260). This proves assertion (1). (2) Although the process t >—> A (p, T ) dW (p) is not a martingale, it has enough properties of a martingale that its expectation is 0, and that its quadratic covariation matrix is given by the expression in (3.262). The reason for all this relies on the equality: f A (p, A) dW(p) = f T A (p, T) dW(p) - f A(p,T) dW(p ), (3.267) Jt Jo Jo combined with the fact that the process t >—► r A (p, T ) dW (p) is a martingale. So we can apply the Ito isometry and its consequences to complete the proof of assertion (2). An alternative way of understanding this reads as follows. Processes of the form s •— * X x (s), s < 0, and t •—* X x (s) ds, 0 ^ t ^ T, consist of Gaussian vectors with known means and variances. For s ^ t we use the representation in (3.266) for A x (s), and for ^ X x (s)ds we employ (3.260). (3) The proof of this assertion follows the same line as the proof of the assertion in (2). (4) If the stochastic variable Z is normally distributed with expectation p and variance v 2 = E [(Z — p) 2 ], then E [e z ] = e fi+ z v . This result is applied to the variable Z = — (y, ^ X x (s) ds ^ to obtain the equality in (3.265). This completes the proof of Proposition 3.92. □ 3.93. Lemma. Let the notation and hypotheses be as in the proposition 3.91 and 3.92. Suppose that the matrix A is invertible. The following equality holds for 0 < t < T: A (P, T ) (P, T )* dp = (T - t)A~ l aa* (A*) -1 - A ( t, T ) A -1 era* (A*) -1 - A -1 a a* (A*) -1 A ( t, T)* - T-t 1 —1 _ ( A*\ — 1 A —1 ^ ( A*\ — 1 + f e~ pA A~ 1 <7<j* (A*)" 1 e“ pA * dp. Jo (3.268) If the invertible matrix A is such that Aaa* = aa*A*, then the following equality is valid for 0 < t < T: r T | A (p, T) aa*A (p, T)* dp 1 = {T- t)A~ l aa* (A*)" 1 - A (t, T ) A~ 1 aa* (A*)" 1 - - (A(t, T)f oa* (A*) -1 206 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales = (t -t- A(t,T) - A (A(t, T)) 2 ) A- 1 a a* (A*) -1 (3.269) Observe that an equality of the form Ago* = go* A* holds whenever A = A* and the matrix 00* is a “function” of A. In particular this is true when d = 1 and A = a is a real number. PROOF. Since A is invertible we have A (p, T) = (I — e ( T A" 4 ) A and so A (p, T) gg* A (p, T)* dp = | T (I - e~^ A ) A~ 1 oo* (A*)- 1 (/ - dp = J (I- e~ pA ) A~ l oo* (A*) -1 (i - e~ pA *^J dp = {T - t)A~ l oo* (A*) -1 - A ( t, T ) A -1 era* (A*) -1 - A~Vct* (A*) -1 A (t, T ) H - T-t (3.270) l —1 _ ( /I * \ — 1 A—( A*\ — 1 + f e- pA A~ l oo* {A*)- 1 e~ pA * dp. Jo The hnal equality in (3.270) proves (3.268). Next we also assume that Ago* = go*A*. Then oo*e ~ pA * = e~ pA oo*, and hence rT-t e- pA A~ 1 GG*{A*)- 1 e- pA * dp Jo "T-t = f e 2pA dp A 1 aa*(A*) ~e Jo = - (/ - e 2{ - T ~ t)A ) A~ 2 oo* (A*)- 1 e ~ pA *. A simple calculation shows l {I- e- 2(T “ tM ) = A(t, T)A - i (A(f, T)f A 2 , (3.271) (3.272) and so the equalities in (3.271) show f T ' e- pA A~ X GG* (A*)' 1 e~ pA * dp = C ' e~ 2pA dpA^aa* (A*)' Jo Jo = ( A(t, T)- 1 - (AT T)f A ) A~ 1 oo* (A*) -1 . (3.273) A combination of (3.270) and (3.273) together with the equality gg* A (t, T)* = A (t, T ) gg* then yields the equality in (3.269), completing the proof of Lemma 3.93. □ Before we discuss the Vasicek model we insert Girsanov’s theorem formulated in a way as we will use it in Theorem 3.101. In fact we will formulate it in a multivariate context. 207 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.94. Theorem. Let { X(t ) : 0 < t < t} be an ltd process satisfying dX(t) = v(t) dt + u(t) dW(t). 0 < t < T. Suppose there exists a process {9(t) : 0 < t < T), with the property that i2 P fl^)l _Jo dt < oo 1, such that the process v(t ) — u(t)9(t) has this property as well. Assume further¬ more that the process t *—» £(t), 0 ^ t < T, defined by £(t) = exp 0(s) dW(s) — ^ ds (3.274) is a V-martingale, which is guaranteed provided E [£(£)] = 1 for 0 < t < T. dP* Define the measure P* such that = £(T). Then t~W*(t) :=W(t)+ f 9{s)ds, t 6 [0, T], Jo is a Brownian motion w.r.t. P* and the process {X(t) : 0 < t < T} has a rep¬ resentation w.r.t. W*{t) given by dX(t) = (v(t) — u(t)9(t)) dt + u(t ) dW*(t). AACSB ACCREDITED Excellent Economics and Business programmes '\&r university of groningen www.rug.nl/feb/education .uHr p W “The perfect start of a successful, international career.” CLICK HERE to discover why both socially and academically the University of Groningen is one of the best places for a student to be 208 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales We shortly show that {£(£): 0 ^ t ^ T) is a diffusion process. Set Y(t) 0(s) dW(s), 0 < t < T, and consider the function f(t,x ) 6 (7 2 ([0, T], defined by f(t,x) = exp ^ J |d(s)| 2 ds^j Then we clearly have that £(£) = f (t,Y(t)). By Ito’s formula we have dE[t ) \9(t)\ 2 E(t) dt - E{t)9(t) dW(t) + i E(t)d(Y,Y) (t) Zj = — \6(t)\ z E(t) dt — E(t)9(t ) dW(t) H—£(£) |d(t)p dt 2 2 = -9(t)E(t)dW(t). Hence, it follows that (3.275) E(t) = £(0) — f 0(s)E(s)dW(s), Jo which in general is a local martingale for which E [£ (7)] < 1. It is a sub¬ martingale, but not necessarily a martingale. If, for 0 < t < T, the expecta¬ tion E [£(£)] = 1, then t i— > £(£), 0 < t < T, is a martingale. If Novikov’s condition, i.e.. if E exp I | \9{t)\ 2 dt j < go is satisfied, then the process {£(£): 0 < t < T} is a martingale. For details on this condition, see Corollary 4.27 in Chapter 4. For more results on (local) exponential martingales see sub¬ section 1.3 of Chapter 4 as well. In section 3 of the same chapter the reader may find some more information on Girsanov’s theorem. In particular, see assertion (4) of Proposition 4.24 and Theorem 4.25. 8.1. The Vasicek model. In this subsection we want to employ the results in Proposition 3.92 with d = 1 to find the bond prices in the Vasicek model. Until now we were always working in the physical probability space (12, T, P). In order to calculate the fair price of a financial instrument one often uses the method of risk-neutral pricing. Through this technique the price of a financial asset is the expectation of its discounted pay-off at the so-called risk-neutral measure Q. The risk-neutral measure is equivalent to the physical measure P. Suppose for example that {S(t)} s>0 is the price of a certain asset at time t. The price of our asset at time t discounted to time 0 is then given by S(t) : = e-^ r(u ) du S(t). As a main property of the risk-neutral measure, the family of discounted prices -iSYt)! is a Q-inartingale. This means that for every s, t J tjs o 0 < s < t, we have E S(t) I = E e~ So du S(t) I = e" So r (“) du S(s) = S(s) , (3.276) where expectations E are with respect to Q. Because of this property, a risk- neutral measure is also called an equivalent martingale measure. Roughly speak¬ ing, the existence of such a measure is equivalent with the no-arbitrage assump¬ tion. We will use this martingale property to price a zero-coupon bond. That is a financial debt instrument that pays the holder a fixed amount named the 209 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales face value at maturity T. For simplicity we take 1 as face value. The price of a zero-coupon bond is then given by the following theorem. 3.95. Theorem. Consider a zero-coupon bond which pays an amount of 1 at maturity T. The price at time t < T is then P(t,T) = E (3.277) Proof. We use the above explained property that the discounted price e -So r ( s '> ds P(t,T) is a martingale, and the trivial fact that P(T,T) = 1, e -$o < s ) ds p{t,T) = E \ e -^ r{s)ds P(T,T) | Tj = E = e-SoWl^E e~^ r ^ ds I e - So r(s) ds I gr (3.278) The bond’s price can thus be written as P(t,T ) = E completes the proof of Theorem 3.95. e - Sf r(s) ds This □ Formula (3.277) is an expression of the bond’s price for an arbitrary chosen interest rate process. We will now apply this to our Vasicek model {r(t)} t>0 . We will investigate three methods that all lead to the same result stated in the following theorem. We follow the approach of Mamon in [94]. For an alternative approach see [114] as well. 3.96. Theorem. Consider a zero-coupon bond which pays an amount of 1 at maturity T. Suppose that under the risk-neutral measure the short rate follows an Ornstein-Uhlenbeck process: dr(t ) = a(b — r(t )) dt + a dW(t). The fair price of the bond at time t ^ T is then given by P(t, T) = g -A(t,T)r(t)+D(t,T) , (3.279) where A(t,T) D(t,T) l — g -a(T-t) and (A(t,T) - T + t)(a 2 b- a 2 / 2) a 2 A(t,Tf a z 4 a (3.280) Equation (3.279) is an affine term structure model. In fact, the bond yield yt(T) is defined as the constant interest rate at which the price of the bond grows to it’s face value, i.e., P{t,T)e yt ^ T ' ) ^ T ~ t ' > = 1. We thus find that y t (T) = zTlfMl = A(t,T)r{t) - D(t,T) ' which is indeed affine in r(t). The yield curve or term structure at time t is the graph (T,y t (T)). 210 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 8.1.1. Bond price implied by the distribution of the short rate. The first method to calculate the bond price is quite straightforward. It calculates the conditional expectation in formula (3.277) by determining the distribution of E |j t T r(s) ds | J t First proof of Theorem 3.96. Because formula (3.277) shows that the bond’s price at time t is conditional on %, we may assume that r(t ) is a pa¬ rameter. Using formula (3.266) (for d = 1, and A = 1) with starting time t we find, for s > t, rls r(t)e —a(s—t) + b( 1 - e“ a(s “ t) ) + J e~ a -a(s-p) adW(p). We want to determine the distribution of r(s > ds conditioned by “J t . Note that because of the Markov property of the Otnstein-Uhlenbeck process (or more generally for diffusion processes: see the equality in (3.249)), this distri¬ bution will only depend on r(t). Let’s start by determining the distribution of ^r(s) ds given T t . This distribution is normal, and essentially speaking this it follows from Proposition 3.92 and Lemma 3.93. First of all from assertion (3) in Proposition 3.92 we get by (3.263) E I r ^ ds % E J r(s) ds | r(f) = A(t, T ) (r(f) — b) + (T — t)b. (3.281) Secondly from (3.264) and (3.269) in Lemma 3.93 we get var | X x (s)ds | X x (t)^j = | cr 2 (A(p,T)) 2 dp = -lT-t-A{t,T)--{A{t,T))‘ (3.282) a* \ 2 The equality in (3.279) of Theorem 3.96 then follows from (3.282) and (3.265) in (4) of Proposition 3.92. □ 8.1.2. Bond price by solving the PDE. A second method that is proposed to calculate the bond’s price in the Vasicek model, is by solving partial differential equations. More precisely, we will derive a PDE for the bond’s price by using martingales. Taking into account the Markov property of the process {r(t)} t>0 (see equality in (3.249)) one can introduce the following variable: P(t,T ) = E = E e -Sfr(s)ds I gr = E \z=r(t) g-St r(s)(z)ds Here r(s), s > t, is the function of r(t) given by r(s) = r(t)e- a(s - t} + b (l - + f e“ a e -h r(s)ds :P(t,T,r(t)). (3.283) e -a(s-p) dW ( p y (3.284) 211 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales We now provide a second proof of Theorem 3.96. Second proof of Theorem 3.96. We will apply Ito’s formula to the function f(t,x ) = e~^o r ( s ') ds P(t i T, x). Then we obtain E e - £ r(s) ds _p ( 0) r(0) ) | gr t j = e - £ W) T, r(t)) — P (0, T, r(0)) f [~r(«)e~ % r(s) ds P (u, T, r(u)) + e~ % r(g)ds — J’ 1 d« Jo + 1 ,-CfrMcfa ^ (u,T,r(u)) (J + T o L 2 J Jo dr{u) - ft r(«) d 2 P(R,T,r(n)) e Jo dr(v,y (a (6 — r(n)) du + cnifT('u)) du. (3.285) Put /(/) = -r(t)e~^ r ^ ds P(t,T,r(t)) + e ~So (/ 6 + dr(u) (a ( b-r(t ))) 2 dr{t) 2 (3.286) American online LIGS University is currently enrolling in the Interactive Online BBA, MBA, MSc, DBA and PhD programs: ► enroll by September 30th, 2014 and ► save up to 16% on the tuition! ► pay in 10 installments/2 years ► Interactive Online education ► visit www.ligsuniversity.com to find out more! Note: LIGS University is not accredited by nationally recognized accrediting agency by the US Secretary of Education. More info here. 212 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales From the equality in (3.285) it follows that the process t >—► ^ f(u) du is a martingale. By Lemma 3.97 below it follows that f(t) = 0 P-almost surely. From (3.286) it then follows that the function P(t,T,x) satisfies the following differential equation: —xP ( t , T, x) + dP ( t , T, x) dt dP(t,T,x) + ax (a{b x )) + a 2 d 2 P (t, T, x) 2 dx 2 From (3.283) and (3.284) it follows that ~ i 1 ~ e_a(T_t) ) P(t, x ) = T ) P (T, x). From (3.288) we easily infer that P (t, T, x) = C(t,T)e~ Ait ’ T)x . = 0. (3.287) (3.288) (3.289) Inserting this expression for P (t, T, x) into (3.287) yields the first order equation —x + 1 dC{t, T) dA(t, T) a C(t, T ) dt dt x — A(t, T) {a(b — x)} + — A(t , T ) 2 = 0. (3.290) Because —1 — ~ + aA(t,T) = 0, the equality in (3.290) implies: (J L (3.291) Since C(T,T) = P (T,T, 0) = 1 from (3.291) we infer C(t,T) = e D( ' t,T ' > and hence P(t,T) = P(t,T,r(t )) = e -Mt,T)r(t)+D(t,T )^ which completes the proof of Theorem 3.96 by employing the PDE as formulated in (3.287). □ The equation in (3.287) is called the PDE for the bond price in the Vasicek model. 3.97. Lemma. Let (f 1, (37) t>0 , P) be filtered probability space, and let the right- continuous adapted process {f(t)} t>0 be such that for some sequence of stopping times (n n ) n6N , which increases to go, the integrals ^ |/(s)| l[i, r „] ds are finite P- almost surely. If the process 1 1 —> u /(s) ds is a local martingale, then f(t) = 0 P -almost surely for almost all t. PROOF. Fix 0 < T < GO. By localizing at stopping times (r^) n e N, r n | go {n —»• go) we may assume that rT E f I/« ds < oo. neN’ n ^ ‘ni (3.292) Otherwise we replace fit) with / (t) l[o, r '](0> an( ^ P rove that / (t) l[o, r '](^) = 0 for all n e N. But then f{t) = 0, by letting n — > go. So we assume that (3.292) 213 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales is satisfied. Then for 0 ^ s < K T we have f f(p) dp + E f f(p) dp | = E f f(p) dp\T s = f f(p) dp. (3.293) Jo Us J LJo J Jo From (3.293) we infer that E f(p) dp\3 r s = 0, P-almost surely, for all 0 < s < t < T. differentiating with respect to t then results in E [fit) | T s ] = 0 P-almost surely for all 0 < s < t < T. But then, by the right-continuity of the process {/(£)}*> 0 if follows that f(s) = limE [f(t) | T s ] = 0, P-almost surely. This completes the proof of Lemma 3.97. □ 8.1.3. Bond prices using forward rates. The third and last method to calcu¬ late a bond’s price in the Vasicek model, is based upon the concept of forward rates. Indeed, in the Heath-Jarrow-Morton pricing paradigm the closed-form of the bond’s price follows directly from the short rate dynamics under the so- called forward measure. Suppose we are at time t. We want to know the rate of interest in the period of time between T\ en 77 with t < T\ < 77. This is called the forward rate for the period between T\ and T 2 and we denote it by f (t, Ti, T 2 ). When the rates between time t and 7\ and between time t and T 2 are known - write R\ and 7? 2 - we must have: e Ri (Ti-t) e /(i,Ti,T 2 )(T a -Ti) = e R 2 (T 2 -t)' Hence, we find for the forward rate n ( , rp rp X _ R 2 {T 2 ~ t) ~ Rl (Tl - t) J -2 -^1 Applying this in our framework of bond prices, R\ and 7? 2 equal the bond yields: Ri = -log P{t,Tf) Ro = — log P (t, Tf) T { -1 ’ T 2 -t such that the forward rate is given by — log P (t, Tf) — log P (t, T\) f (t, Ti, T 2 ) = T 2 ~ Ti When Ti and T 2 come infinitesimally close to each other, we obtain a so-called instantaneous forward rate. The instantaneous forward rate at time T > t is log P(t,T)~ log T(M') d log P {t, t') , fit,T) = -Km T -V dt 1 I t’=T Solving this partial differential equation for Pit, T) on [t,T] we find immedi¬ ately that Pit,T) = e~^ nt ’ s)ds . (3.294) Later on we will see that the link between the instantaneous forward rate and the short rate is the so-called forward measure. In the sequel, we will need two properties of conditional expectations under change of measure. These results can be found in [ 32 ] . In the following theorems P is a probability measure on a 214 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales cr-algebra T, the probability measure Q « P is such that = Z. Furthermore 9 is a sub-u-algebra of 3\ The symbol E denotes expectation w.r.t. P, while E^ stands for expectation w.r.t. Q. 3.98. Theorem. In the notation of above, it holds that dQ \ 5 dF L E [Z | 9] . Proof. Take an arbitrary B e 9- We need to show that Q(B) = E [E [Z | 9] 1 B ] ■ Indeed: E [E [Z | 9] 1 B ] = E [E [Z1 b | 9]] = E [Z1 B ] = Q(B). This completes the proof of Theorem 3.98. □ 3.99. Theorem. For any T -measurable random variable X : E [Z | 9] E Q [X | 9] = E [ZX | 9] • PROOF. Let Y = E [Z | 9]- Take B e 9 arbitrary, then: E q [1 b E [ZX | 9]] = E [TIbE [ZX | 9]] = E [E [Y1 b ZX IS]] = E [Y1 b ZX] = E q [Y1 b X] = E q [E q [l b yx | 9 ]] = e q [i b e q [yx | g]]. In the first step we used that 1 #E [ZX | 9] is 9-mesurable. Hence we could apply Theorem 3.98 which tells us that dQ | g = YdF | . Because the previous reasoning holds for all B e 9 we must have: E [ZX | 9] = E q [YX | 9] = YE q [X | 9] what proves the claim in Theorem 3.99. □ As well as the economic term forward rates, we introduce the concept of a numeraire. A numeraire is a tradeable economic security in terms of which the relative prices of other assets can be expressed. This allows us not only to compare different financial instruments at a certain moment, it makes it also possible to compare the prices of assets at different times. A typical example of a numeraire is money. The random variable M (t) = e^o r ( s ) ds represents the value at time t of an asset which was invested in the money market at time 0 with value 1. Recall that in accordance with the definition of a risk-neutral measure Q , the price of an asset relative to the money market is a martingale. In our new notation the expressions in (3.276) become: E S(t) , „ ] = E _M{t) 1 s _ e -ft »■(«)*.£(£) I j 1 = e -ft r(n)du S ^ S(s) M(s )’ with 0 < s < t. We say that Q is an equivalent martingale measure for the numeraire {M(t)} t>0 . Let N(t) be the price at time t of another traded asset. 215 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Suppose that Q* is an equivalent martingale measure for {N(t)} t>0 , i.e. for all 0 < s < t: E* S(t) m gw N(‘Y We can also define this measure on the basis of the Radon-Nikodym derivative of Q* w.r.t. Q. 3.100. Theorem. Suppose that Q is an equivalent martingale measure for the numeraire {M(t)} t>0 . Let Q* be an absolutely continuous measure w.r.t. Q defined by the Radon-Nikodym derivative: dQ* , _ M(0) N(t) ~dQ M{t) N(0) ’ (3.295) where N(t ) > 0 is the price at time t of a particular asset. Then Q* is an equivalent martingale measure for {N{t) : t ^ 0}. Proof. Denote expectations w.r.t. Q by E and w.r.t. Q* by E*. Let S(t) be the price of an asset at time t ^ 0 and assume S(t) e L 2 (Q, T t ,Q) n L 2 (0,3y, Q*). For t ^ s ^ 0 we find using Theorem 3.99 E* = E _N(t) 1 s _ M(Q)N(t) S(t) M(t)N( 0) N(t) J. /E M(0)N(t) M( 0) m E Sit) M(t ) To M(t)N( 0) N(0) M(s) _ S(s) M( 0) N(s) ~ N(s)' The proof of Theorem 3.100 is complete now. □ A cate-Lucent www.alcatel-lucent.com/careers What if you could build your future and create the future? One generation’s transformation is the next’s status quo. In the near future, people may soon think it’s strange that devices ever had to be “plugged in.” To obtain that status, there needs to be “The Shift". 216 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Note that the measures Q and Q* are equivalent because of the strictly posi¬ tiveness of the Radon-Nikodym derivative. We already mention the following theorem which transforms the dynamics of a process under Q to a process under Q*. 3.101. Theorem. LetQ be an equivalent martingale measure for {M(t)} t>0 and let Q* be defined by equation (3.295). Assume {X(t) : 0 < t < t) a diffusion process with dynamics under Q dX(t ) = b (t, X(t)) dt + a (; t , X(t)) dW(t). Let also M(t ) and N(t ) have dynamics under Q given by dM(t) = m,M dt + <jm dW(t), dN(t ) = dt + ctn dW(t). Then the dynamics of {X (t) : 0 < t < t} under Q* is given by dX(t) = b(t,u) dt — a (t, u) ~ dt + a (t, uf) dW*(t), where w ’ (t) = w(t)+ i(wr)-m) ds - Proof. It is clear that we want to apply Girsanov’s Theorem 3.94. But then we need to know how 0 (t,u ) in expression (3.274) looks like. From expression (3.275) we know that dT t = -9(t,-)T t dW(t). (3.296) On the other hand: _ M(0) / N(t) \ _ Af (0) dN(t)M(t) - N(t)dM(t) t ~ m \W)) ~ W) M(t) 2 = jy(0 jMfff 2 (( mNdt + a N dW(t)) M(t) - N(t ) ( m M dt + a M dW(t ))). (3.297) Because {T f : 0<t<T}isa martingale we must have that the coefficient of dt is 0, hence dlt Miff) N(0)M(t ) 2 &n cm \ (a N dW(t)M(t ) - N(t)a M dW(t )) W)-W))™ Comparing this with (3.296) we have that a/. \ c M 0 {t, UJ) Cn M(t ) N(t )' Finally applying Girsanov’s Theorem 3.94 to this we have cm cn dX (t) = b ( t , oj) dt — a ( t , u>) M(t) N(t ) dt + a (t,uj) dW* ( t ), 217 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales with W*(t) = W(t) + r ( % Jo \M(s) <JN M{s) N(s ) Altogether this completes the proof of Theorem 3.101. ds. □ In order to make the link between the short rate r(t) and the instantaneous forward rate f(t,T ), we introduce a new measure Q T . Suppose again that Q is the risk neutral measure w.r.t. the money market and E the expectation w.r.t. Q. 3.102. Definition. Take T ^ 0. The forward measure Q T is defined on by setting T r : = dQ T _ M( 0) ~dQ ~ M(T) P(T, T)P (0, T) = e“^ r(s)ds P(0,T), where M(t) = e& r(s)ds , and P (t, T)= E e - if r (») ds By the previous theorem we conclude that Q T is an equivalent martingale mea¬ sure which has a bond with maturity T as numeraire. For t < T we can easily calculate T t as follows: T t := E [r T | T t ] = M( 0) ¥{0/r) E P(t,T) P (0, T) M(t) = P -5o r ( s ) ds P(d]T) | M(T) 1 * PfaT) P (0,T) ’ where we used in the second to last equality that Q has {M(t)} t>0 as numeraire. Now we have all theoretical background information to formulate the third proof of Theorem 3.96. Third proof of Theorem 3.96. Denote as before the expectation w.r.t. Q by E and the expectation w.r.t. Q T by E T . We have by Theorem 3.99 that for any T^-measurable random variable X and t^T E r [Xj%] = T-'E [XT t | T t ] = E r T , x % r i = E M(t)P(T, T) ~M(TWKT) X = E X- ~ if r(s) ds ,e P(t,T) We want to express the forward rate in terms of the short rate. We got a formula for the bonds price in function of both of them. Differentiating expression (16) towards T gives dP (t, T ) dT = E —r(T)e E r [r(T) | Jf r(s) ds | ^ ?t]P(t,T). E r [~r(T)P(t,T) |T t ] (3.298) 218 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales In the second step we used the above reasoning with X = —r(T)P(t,T). Dif¬ ferentiating now formula (3.294) with respect to T gives dP (; t , T) dT (3.299) = a Comparing (3.298) en (3.299) we get the link between short rate and forward rate f(t,T)=E T [r(T)\%]. (3.300) Considering the right hand side of (3.300) we will need to describe the dynamics of r(t) under Q T . Applying Ito’s formula on f(t,x ) = ds we immediately find that dM(t ) = r(t)M(t ) dt. In the notation of Theorem 3.101 we thus have cm = 0. If we then apply this theorem with X(t) = r(t), Q* = Q T , a (t,X(t)) = a, b (t, X(t)) = a (b — r(t )) and cpy = — crA (t, T ) P (t, T) we obtain: dr(t) = (ab — a 2 A (t, T ) — ar(t )) dt + a dW T {t ) b - (1 — e- a(T - f) ) - rtj dt + a dW T (t ) where W T (t) is the Q r -Brownian motion defined by W T {t) = W{t) + a f A(s,T ) ds. Jo Expression (3.301) resembles an ordinary Vasicek process, except that the term ^2 b -- (l — e _a ( T_t D does depend upon t and is thus not a constant. However, cr v ' we will use a similar reasoning as in the classical situation to solve the SDE for r(t) on the interval [t,T]. First, we apply Ito’s formula on g(t,x ) = e at x: 1 _ p -a(T-t) d (e at r(t)) = abe at dt - a 2 e at dt + ae at dW T (t ) (3.301) = abe at dt- — (e at - e~ a{T ~ 2t) ) dt + ae at dW T {t). Integrating from t to T gives e aT r{T) - e at r{t) = b(e a e as ds - ^ | (e as - e - a{T ~ 2s) ) ds + a | *) C7 a a 2 i V — ( / O^, V 2 a -a( T ~ 2 ‘) e as dW T (s ) - a(T -20) +CJ | e as dW T (s) ) +C7 l eaSdWT = b (e aT - e at ) -- ( e aT - 2e at + e v ' 2 a 2 Thus we have that r(T) = r(t)e- a(r “ t) + b (l - e -“( r -*)) - ^ (l - 2e“ a(T - t) + e -2«(r-0) ZiLL rjn | e~ a{T ~ s) dW T {s). + a 219 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales And hence, f(t,s) = E s [r(s )\%] 2 = r(t)e~ a ^ +b( 1 - e - a(s - 4) ) - G — (l - 2 e -“ (s -* ) + e" 20 ^) \ J a 2 \ / = r(t)e~ a ^ + (b- — 2 ) (1 - e- a(s - t} ) + -A ( e -<W-9 _ e -M*-t )) _ Integrating results in f E> I * = IT f 1 - + ( 6 -1?) ( r - * - + a 2 (l_ e -a(T- t) )_i_(i_ e - 2cl (T- t )) (: T-t-A(t,T )) + '—A(t,TY r(t)A(t,T ) + 6 cr = r(t)A(t,T) - D(t,T). (3.302) Reminding formula (3.294) and formula (3.300) we find again that p(t : T) = e~ A ( t ’ T ') rt+D ( t ’ T \ This completes the third proof of Theorem 3.96. □ In the past four years we have drilled * 81,000 km A That's more than twice around the world. Whn am wp? fHSHHHH P We are the world's leading oilfield services company. Working 1 globally—often in remote and challenging locations—we invent, design, engineer, manufacture, apply, and maintain technology to help customers find and produce oil and gas safely. Who are we looking for? We offer countless opportunities in the following domains: ■ Engineering, Research, and Operations ^ ■ Geoscience and Petrotechnical ■ Commercial and Business A ^ If you are a self-motivated graduate looking for a dynamic career, apply to join our team. What will you be? careers.slb.com Schlumberger 220 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 9. A version of Fernique’s theorem The following theorem is due to Fernique. We follow the proof of H.H. Kuo [77], 3.103. Theorem. Let (f2,T, P) be a probability space and let X : fl — » be a Gaussian vector with mean zero. Put 1 a = sup U,V>0 log P( \x P v ) P(l \x\ > u ) (3.303) Then a > 0 and E ( exp I -rj |X| 2 ) 1 < go for r/ < a. (3.304) For the proof we shall need two lemmas. The first one contains the main idea. 3.104. Lemma. Let (fl,T, P) and X be as in Theorem 3.103. Let s > 0 be such that P (|X| < s) > 0 and fix t > s. Then P (\X\ > t) ^ ( P (\X\ >{t- s)/V 2) F{\X\ < s) ^ \ P(|X| ^ 7) 2 (3.305) PROOF. Let (fl ® fl, T ® T, P ® P) be the tensor product space of (fl, T, P) with itself and define X i: i = 1, 2, by Xi(u>i,uj 2 ) = X(ujf). Then the variables Xi and X 2 are independent with respect P ® P and their P ® P-distribution coincides with the P-distribution of X. We shall prove Lemma 3.104. for s = v and t = u \/2 + v. Since the vector ( Xi,X 2 ) is Gaussian with respect to P®P and since the components of X\ — X 2 are uncorrelated with the components of Xi + X 2 (with respect to the probability P ® P), it follows that the vectors Xi — X 2 and Xi + X 2 are independent. Notice that $XdP = JXidP® P = J X 2 dP ® P = 0 and that the covariance matrices of X, of (X 1 — X 2 ) /y'd and of (X\ + X 2 ) / y'2 all coincide. It follows that the joint distributions of (X\. X 2 ) and of ( — 1 are the same as well. Hence the following (in- V2 ’ V2 )equalities are now self-explanatory: P (|X| ^ u) P > u\l 2 + = P®P(|Xi| ^ v) x P ® P 1 > u\! 2 + v'j = P®P^|Xi|<u and \X 2 \ > u\[2 + v'j = P® P ^|Xi — X 2 \ < vV2 and \Xi + X 2 \ > 2u + vV^j < P®P ^||Xi| — |X 2 || ua/ 2 and |Xi| + \X 2 \ > 2u + vV^j < P®P(|Xi| > u and |X 2 | > u) = P (|X| > uf . (3.306) Inequality (3.305) in Lemma 3.104 follows from (3.306). □ 221 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.105. Lemma. Let (fl, T, P) and X be as in Theorem 3.103. Let v > 0 be such that P (|X| y v) > 0 and fix £ e N and fix u > 0. Then the following inequality is valid: P (j x\ \>u(V2)' + v( (V2) ■r 1 ) 1 (a/2 + 1) L, mi \X\ > u) P(l \x\ <0 " 1 \F(\X y v) 2 * (3.307) PROOF. For £ = 0 this assertion is trivial and for £ = 1 it is the same as inequality (3.305) in Lemma 3.104. Next suppose that (3.307) is already established for £. We are going to prove (3.307) with £ + 1 replacing £. Again we invoke inequality (3.305) to obtain P (|A'| > u (V2 )' +1 + v ((V2)' + ‘ - l) (V 2 + 1 )) P(|X| « v) [P(\X\> (« (V2)' +1 + v ((V2)' +1 - l) (V2 + 1) - „) /X) * ( /P (|A| > u (V 2 )' + t, ((V 2 )' - l) (V 2 + 1 )) 7 PPM ) (induction hypothesis) P(| x\ > u ) P(|X| /A 2 i+1 The inequality in (3.308) completes the proof of Lemma 3.105. (3.308) □ Proof of Theorem 3.103. If X m 0, then there is nothing to prove. So suppose X 7 ^ 0 and choose strictly positive real numbers u and v for which P(|X| > u) P(|X| y v) < 1. Put a(u,v ) 1 (U + V ) 2 log P( \x\ y v ) P(|X| > 0 and P(|X| y v) f 2 (1 + V2) 2 / P (|A| > u) \ 2 (u + v) 2 Then a(u,v ) > 0 and 0(u,v) = P(|A| ^ e)exp (— ^a(u,v)v 2 (l + V2) 2 ) < 1- For s ^ u choose t e N in such a way that 222 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Then 2* > (s + v (l + a/ 2 ))‘ 2 (u + v ) 2 > (1 +V2 y 2 (u + v) z 2 (u + v ) 2 + and hence P(|X| > s) < P > u + v (inequality (3.307) in Lemma 3.105) 2 -1 2 + 1 < P(|X| ^ v) P(|X| > u) P(|X| < v) < f3(u,v)ex p ( — -a(u, v)s 2 j < exp ( — -a(u,v)s 2 (3.309) If 0 < rj < a, then we choose u, v > 0 in such a way that a > a(u, v ) > rj. Then, for s ^ u, P(|X| > s) < exp (— ^a(u, u)s 2 ). Consequently, we get from (3.309): IE ( exp ( -rj \X\ ) :\X\ >u fH P ( exp ( ^ |X | 2 ) > f, \X\ > u ) di (substitute £ = exp (|r^s 2 )) < exp P (|X| > u) + r) r P (|X| > s ) exp (^V s2 ^j s ds + exp P(|X| > u) + rj J 1 .2 \ , V < exp \-rju + - . 2 J a(u,v)-rj a(u,v ) (\ < -7 -\-exp -rju‘ a(u, v) — rj \2 exp ( — - (a(u,v) — rj) s 2 ) sds 1 exp ( —- ( a{u , v ) — ?y) u 2 From (3.310) we infer E (exp (jxM 3 )) < (/+ Inequality (3.311) yields the desired result in Theorem 3.103. (3.310) (3.311) □ 10. Miscellaneous We begin this section with the Doob’s optional stopping property for discrete time submartingales. Let { X{n ) : n e N} be a submartingale relative to the filtration {T n :neN}. Here the random variables X(n) are defined on a prob¬ ability space (f2,T, P). The following result was used in inequality (3.164), the basic step for the continuous time version of the following proposition. 223 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 3.106. Proposition. Let r be a stopping time. The process {X(min(n, r)) :neN} is a submartingale with respect to the filtration {T n :neN} as well as with respect to the filtration {5F m i n (n,r) '■ n^N}. / i Maastricht University Join the best at the Maastricht University School of Business and Economics! gjpj* • 33 rd place Financial Times worldwide ranking: MSc International Business • 1 st place: MSc International Business • 1 st place: MSc Financial Economics • 2 nd place: MSc Management of Learning • 2 nd place: MSc Economics • 2 nd place: MSc Econometrics and Operations Research • 2 nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier ‘Beste Studies' ranking 2012; Financial Times Global Masters in Management ranking 2012 Maastricht University is the best specialist university in the Netherlands (Elsevier) Master's Open Day: 22 February 2014 www.mastersopenday.nl j 224 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales PROOF. Let m and n be natural numbers with m < n and let A be a member of 3 m . Then we have E (X (min(n, r)) 1^) — E (X (min(m, r)) 1^) n = {E(X(min(fc,r)) In) - E (X (min(fc - l,r)) U)} k=m +1 n = 2 {E((X(min(fc,r))-X(min(A:-l,r)))U n{T>fc} )} k=m+l n = E {E ((X (min(A:, r)) - X (min(fc - 1, r))) 1 A n{r>k}) \ 3 k ~ 1 } k=m+l (the event .A n {r ^ A:} belongs to 3 k ~i for k ^ m + 1, and the variable X(/c — 1) is Tfc-i-measurable) n = E((E(X(/c)|T fc _ 1 )-X(/c-l))l An{T ^ } ) fc=m+l (submartingale property of the process (X(A;) : k e N}) n > 2 E (0 x 1 An{r>k}) = 0. (3.312) fc=m+1 The inequality in (3.312) proves that the process {X(min(A;, r)) : k e N} is a sub¬ martingale for the filtration : k e N}. Since the cr-field T m i n (^ T is contained in the a-field 3T, k e N, it also follows that the process {X(min(fc,r)) : k e N} is also a submartingale with respect to the filtration (Tmin^) : k e N} because we have E (X(min(m + l,r)) | T min ( m)r )) = E (E (X(m + 1) | 3 m ) | 3m in(m>T )) > E (E (X(min(m + l,r)) | T rn ) | T min(m , r) ) (employ (3.312)) > E (X(min (m,r)) | 3m in ( m;T )) = X(min (m,r)). (3.313) The inequalities (3.312) and (3.313) together prove the results in Theorem 3.106. □ Next we prove Doob’s maximal inequality for martingales. 3.107. Proposition. Let { M(n ) : n e N} be a martingale. Put M(n)* = max|M(n)|. k^n The following inequalities are valid: P lM{n )* ^ A] < ^-E [\M(n)\ : M(n)* ^ A]; (3.314) A 225 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales P \M(ri)* ^ A] < [|M(n)| 2 : M{n)* ^ X] . (3.315) Let {M(t) : t ^ 0} be a continuous time martingale that is right continuous and possesses left limits. Put M(t)* = sup n<(!< JM(s)|. Aqain inequalities like (3.314) and (3.315) are true: P {M(ty ^ A} < \E{\M(t)\ : M{ty > A}; (3.316) A P { M(ty ^ A} ^ {|M(t)| 2 : M(ty ^ A} . (3.317) Proof. We begin by establishing inequality (3.314). Define the events A k , l^k^n,b y A o = {\M(0)\>X}, A k = {\M(j)\ < A,0 ^ j k - 1 , \M{k)\ > X} , 1 < k < n. Then (J^ =0 A k = { M(n )* ^ A}, A k n An = 0, for k =|= £, 1 ^ k, i ^ n, and A k is Tfc-measurable for 1 < k < n. Moreover on the event A k the inequality \M{k)\ ^ A is valid. From the martingale property it then follows that: P(M(«)* 3= A) = £>(40 =S i|]E(UJM(i)|) k =0 A k =0 = i f; E (u, |E (M(n) T4I) ^ k= 0 = li]E(|E(n,M(n)|:Ji)|) ^ k= 0 « i 2 E (E (n,|M(n)| |%)) ^ k= 0 = i 2 E (U, \M(n)\) = 1 e (\M(n)\ : M(n)' 3= A). (3.318) /c=0 Notice that inequality (3.318) is the same as (3.314). The proof of (3.315) goes along the same lines. The fact is used that the process {|M(n)| 2 : neN} constitutes a submartingale. The details read as follows. The events A k , 1 ^ k < n, are defined as in the proof of (3.314). The argument in (3.318) is adapted as below: n n P (M(»)* > A) - 2 P (A) s T5 2 E (n„ IM(fc)| 2 ) k =0 k =0 -i n -i n =5 P 2 E (n»E {\M(nf I Si)) < (U* l M (")| 2 ) /c—0 fc—0 1 71 1 = T5 2 E (u, |Af(„)| 2 ) = -E (|A/(„)| 2 : M(„)> 3= A). k =0 (3.319) 226 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Again we notice that (3.319) is the same as (3.315). The inequalities in (3.316) and (3.317) are based on a time discretization of the martingale { M(t ) : t F 0}. Therefore we write N(j) := M (j2~ n t) and we notice that {N(j) : j e N} is a martingale for the filtration : j e N}. From (3.314) we obtain the inequality: P (rnM |JV(j)l » a) « 1e (|M(f)| : M(t)‘ S> A). (3.320) Inequality (3.317) is obtained from (3.320) upon letting tend n to go. The proof of (3.317) follows in the same manner from (3.315). □ We continue with a proof of the (DL)-property of martingales. More precisely we shall prove the following proposition. 3.108. Proposition. Let {M(s) : s ^ 0} be a right continuous martingale on the probability space (O, T, P). Fix t ^ 0. Then the collection of random vari¬ ables {. M(t ) : 0 < r ^ t, r stopping time } is uniformly integrable. PROOF. Fix a stopping time 0 ^ r ^ t and write r n = min(2 _n [2 n t\,t). Then 0 ^ r n ^ t and every r n is a stopping time. Moreover r n [ r if n tends to oo. Since the pair (. M(r n ),M(t )) is a martingale for the pair of cr-fields (9y n ,Tt) (Use Proposition 3.106 for martingales), the pair (|M (r n )|, \M(t)\) is a submartingale with respect to the same pair of cr-fields. As a consequence we obtain: E(|M(r)| : |M(r)| ^ A) = E (liminf |M(r n )l { | M ( T „)>A}|) (Fatou’s lemma) lirninfE (\M(r n )\ l{\M{r n )\>\}) (submartingale property) < liminf E (E (\M(t)\ \ T Tn ) l{| M (r n )|^A}) n^> oo v v 1 7 7 = liminf E (\M(t)\ 1{|m(t„)|^a}) n^> oo = liminf E (|M(t)| : |M(r n )| ^ A) n —>oo < E : \M(t)\ > A). (3.321) This proves Proposition 3.108. □ Remark. In the proof of Proposition 3.108. we did use a discrete approximation of a stopping time. However we could have avoided this and consider directly the pair (M(r), M(t)). From Proposition 3.107 we see that this pair is a martingale with respect to the pair of cr-fields (T r , T t ). This will then imply inequality (3.321) with r replacing r n . On the other hand the discrete approximation of stopping times as performed in the proof of Proposition 3.108 is kind of 227 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales a standard procedure for passing from discrete time valued stopping times to continuous time valued stopping times. This is a good reason to insert this kind of argument. The main result of Section 3 of this chapter says that linear operators in Cq(E ) which maximally solve the martingale problem are generators of Feller semi¬ groups and conversely. In the sequel we want to verify the claim in the example of Section 3. Its statement is correct, but its proof is erroneous. Example 3.49 in Section 3 reads as follows. 3.109. Example. Let L 0 be an unbounded generator of a Feller semigroup in Cq(E) and let /i& and Z 4 , 1 ^ k ^ n, be finite (signed) Borel measures on E. Define the operator L as follows: =n y k =1 ^ Lwf = Lof, feD (Lppj . Then the martingale problem is uniquely solvable for Lp^. In fact let {(a T, P*), (X(t) £)} be the strong Markov process associated to the Feller semigroup generated by L 0 . Then P = P.,. solves the martingale problem (a) For every / e D(L^p) the process /(*(*)) - /(X(0)) - f L p ,pf(X(s))ds, t > 0, Jo is a P-martingale; (b) P(X(0) = x) = 1, uniquely. In particular we may take E = [0,1], Lof = \ f" , O(t„) = {/EC 2 [0.1]:/'(0)=/'(l)=0}, Hk (I) = u k = 0, 0 ^ a,k < Pk ^ 1, 1 < k ^ n. Then L 0 generates the Feller semigroup of reflected Brownian motion: see Liggett [86], Example 5.8, p. 45. For the operator Lp t p the martingale problem is uniquely (but not maximally uniquely) solvable. However it does not generate a Feller semigroup. From the result in Theorem 3.45 this can be seen as follows. Define the func¬ tionals A j : D(Lq) —*■ C, 1 ^ j ^ n, as follows: Lj(f) = J Lofcl+tj - J./Vizq, l^j^n. We may and do suppose that the functionals A ? , 1 p j P n, are linearly independent and that their linear span does not contain linear combinations of Dirac measures. The latter implies that, for every xo e E and for every function u e D(L 0 ), the convex subsets D (Li) n {{g e Cq(E) : Re g ^ Re g(x o)} + u} and D (£„>-) 228 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales D (Li) n {{h e Co(E) : Re h ^ Re h(xo)} + u} are non-void. The latter follows from a Hahn-Banach argument. Hopefully, it will also imply that the quantities in (3.322) and (3.323) coincide. Since D(Lq) forms a core for L 0 we may choose functions u1 < k < n, such that A j(uk) = Sjk and such that every Uk- ; 1 A k y; n is in the vector of the two spaces {ue D(Ll) : R(l)u e D(U)} and {u e D(L 2 0 ) : R(2)u e D(Li)} . As operator L\ we take Li = and for T we take Tf = f e D(L q ). > Apply now REDEFINE YOUR FUTURE AXA GLOBAL GRADUATE PROGRAM 2015 redefining /standards Qr 229 ^0 Click on the ad to read more Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales The remainder of this section is devoted to the proof of the following result. Whenever appropriate we write R(X) for the operator (XI — L 0 ) _1 . 3.110. Theorem. Let L 0 be the generator of a Feller semigroup in Cq{E ) and let Li and T be linear operators with the following properties: the operator I — T has domain D(L 0 ) and range D(Li), L\ verifies the maximum principle, the vector sum of the spaces R(I — T ) and R{Li{I — T)) is dense in Co(E), and the operator L\{I — T) — (/ — T)L 0 can be considered as a continuous linear operator in the domain of L 0 . More precisely, it is assumed that limsup 1(^(7 - T) - (/ - T)L 0 ) R(A)|| < 1. A—»oo Then there exists at most one linear extension L of the operator L\ for which LT is bounded and that generates a Feller semigroup. In particular, if the martingale problem is solvable for L\, then it is uniquely solvable for L\. Before we actually prove this result we like to make some comments. In order to have existence and uniqueness for the extension L on R(T) it suffices that for every v e R(T) and for every xq e E the following two expressions are equal: lim inf jRe L 1 f(x 0 ) : inf Re (f(y) - v(y)) > Re (f(x Q ) - v(x Q )) - e 40 feD(Li) ( yeE (3.322) lim sup \ Re L 1 f{x 0 ) : sup Re (f(y) - v(y)) < Re (f(x 0 ) - u(x 0 )) + e £ 1° feD(L i) f yeE (3.323) This common value is then by definition Re L 2 v(x 0 ). The value of L 2 v(x 0 ) is then given by [L 2 v] (x 0 ) = Re [. L 2 v] (x 0 ) — iRe [L 2 (iv)] (a; 0 ) for v e R(T). Let A j, 1 F .j A n, be as in the example of section 1. For every xo 6 E and for every 1 ^ k ^ n there exist functions gk and hk e D(L 0 ) with the following properties: A e (g k ) = A £ (h k ) = S k/ , Re g k (x) ^ Re g k (x 0 ) and Re h k (x 0 ) ^ h k (x) for all x e E and Re Li {h k — g k ) (xo) = 0. It then readily follows that the two expressions in (3.322) and (3.323) are equal for functions v in the linear span of Mi,... ,u n . Notice that the function Re h k attains its minimum at x 0 and that the function Re g k attains its maximum at x 0 . In order to define [L±u k \ (a;o) we choose functions g k and h k with A e(g k ) = Re A e(h k ) = —S k / in such a way that the function Re g k attains its maximum at xq and that the function Re h k attains its minimum at the same point xq . Moreover we may and do suppose that Re Li (g k — h k ) (xo) = 0. The value [L 2 u k ] (xo) is then given by [L 2 u k ] (x 0 ) = [L x (g k + u k )] (x 0 ) = [L x (h k + u k )] (x 0 ). PROOF of Theorem 3.110. Let L be any linear operator which extends Li and that has the property that its domain D contains R(T) = TD(Lq). We also suppose that L verifies the maximum principle. Let L\ be the restriction of L to R(I — T ) and let L 2 be the operator L confined to R(T). We shall prove that the operator Li has a unique extension that generates a Feller semigroup. We start with the construction of a family of kind of intertwining operators 230 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales {(A) : A > 0 and large}. This is done as follows. The symbol R( A) is always used to denote the operator R(X) = (XI — L 0 ) _1 . Define the operator V by V = Li(I — T) - (I - T)L 0 (3.324) and define the operator V(A), A > ||L 2 T|| (3 , via the equality V(X) = X(XI - L 2 T)~ l V. (3.325) Then we have: (XI - Li) (I-T) + (XI - L 2 ) = (i — T) (XI -Lo- V(X)) . (3.326) A An equivalent form of (3.326) is the equality (XI - U) (I -T) + (XI - L 2 ) T V -f = (I — T) ((XI - L 0 ) - D(A)) (3.327) = (I-T) (XI - L 0 ) -(I- T)V(X) = (I -T) (I - V (X)R(X)) (XI - L 0 ) . Next we shall prove that the martingale problem is solvable for L\. We do this by showing that the operator L\ extends to a generator L of a Feller semigroup. For large positive lambda we define the operators G( A) in Cq(E) as follows. For / of the form f = (I — T)g, with g = (I — V(X)R(X))(XI — Lo)h, we write T) and if the function / is of the form / = (XI — L\) (I — T)g we write G(X)f = G(X)(XI - Li)(I - T)g = (I - T)g. (3.329) h. (3.328) G(X)f = G(X)(I — T)g = ll-T + T If (XI — Li) (I — T)gi = (I — T)g 2 , then, since I — T is mapping attaining values in the domain of Li, we see that (I — T)g\ — (I — T)g 2 belongs to D(Li) and hence the following identities are mutually equivalent (we write 32 = (I ~ V(X)R(X)) (XI — Lq) h 2 ): (I ~ T) gi (I - T)gi -(I- T)h 2 (XI — Li) ((I — T)gi — (I — T)h 2 ) (I ~ T)g 2 (I-T) (I -V(X)R(X)) (XI - L q ) h 2 (XI - Li) (I - T)h 2 + (XI - L 2 ) T^-h 2 A I-T + T^pj h 2 - (XI - Ll )T^h 2 , (A/-L x ) + h 2 - (XI -LJl^I-T + T (XI -h) (i-T + T V(A) A y( a ) A ^ 2 ! ho. (3.330) 231 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Since g 2 = (I — V(X)R(X))(XI — L^)h 2 it follows that (a/ - Z) ((/ - t) (< h - h 2 ) - t i-t-h 2 ) (A I - L x ) ((/ - T) 9l -(I- T)h 2 ) - (XI - L 2 ) T^h 2 = (/ - T)g 2 - (A I - L x ) (I - T)h 2 - (A I - L 2 ) = (I - T) (I - V(A)i2(A)) (A/ - L 0 ) h 2 - (A/ - L x ) (I - T)h 2 - (XI - L 2 )T^h 2 = (XI - L x ) (I - T)h 2 + (XI - L 2 ) T^-h 2 - (XI - L x ) (I - T)h 2 X - (XI - L 2 ) T^XX)-h 2 = 0. (3.331) A Since the operator L verifies the maximum principle, it is dissipative, and so the zero space of XI — L is trivial. We conclude from (3.331) the identity TV(X)R(X)h 2 = (I — T)gi~(I — T)h 2 and so the function TV(X)R(X)h 2 belongs to D(L X ). Hence it follows that (3.330) is satisfied and consequently that the operator G( A) is well-defined. Next we pick h x and h 2 in the domain of L 0 and we write f = X(I- T ) (XI -L 0 - V(X)) h 2 + (XI - L x ) (I - T ) (h x - Xh 2 ). (3.332) A calculation will yield the following identities: G(X)f = (I-T)h x +TV (X)h 2 - XG(X)f — f = L X (I — T)h x + L 2 TV(X)h 2 = L (G(X)f ). (3.333) Consequently we get ^A I — L^j G(X)f = /, for / of the form (3.332). Since we know ||C(A)i?(A)|| /3 < 1 and since, by assumption the subspace R(I — T) + R(L X (I — T)) is dense in Co(E), it follows that the range R(XI — L ) is dense for A > 0, A large. Since the operator L satisfies the maximum principle and since L = L X (I — T) + L 2 T it follows that the operator L that assigns to G(X)f the function XG(X)f - f, f e R(I - T) + R(L X (I - T)), is well defined and satisfies the maximum principle. Below we shall show that the family {G(A) : A > 0, A large} is a resolvent family indeed: see (3.337). The closure of its graph contains the graph {((/ - T)h x + Th 2 , L\(I - T)h x + L 2 Th 2 ) :h u h 2 e D(L 0 )}. Denote the operator with graph {( G(X)f , XG(X)f — f) : f e Co(E)} again by L. From the previous remarks it follows that the operator L verifies the maximum principle, (XI — L)G(X)f = f for / e Cq(E) and that it is densely defined. The latter follows because its domain contains all vectors of the form (I - T)h + L X (I — T)/ 2 = (I-T) (f x + (I - T)L X (I - T)f 2 + TL 0 f 2 ) + TVf 2 . 232 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales From a general argument it then follows that the operator L is the generator of a Feller semigroup: for more details see [Ml], Theorem 2.2 page 14. Next let hi and h 2 belong to D(L 0 ). Then we have A ||G(A) ((/ - T)h, + (XI — L,) (I — T)h 2 ) = A ||G(A)(7 - T)h, + (I- T)M„ (the operator L is dissipative) | (a/ - L) (G(X)(I - T)hi + (/ - T)h 2 ) \\(I-T)hi + (\I-Li)(I-T)h 2 \\ x . (3.334) Since the vector sum of the spaces R(I — T) and R(L\(I — T)) is dense it follows from (3.334) that the operator G( A) extends as a continuous linear operator to all of Cq(E). Moreover it is dissipative in the sense that A||G(A)||<1. (3.335) Next we prove that the operator G( A) is positive in the sense that / ^ 0, / e Co(E), implies G(X)f ^ 0. So let / e Cq(E) be non-negative. There exist sequences of functions ( g n ) and ( h n ) in the space D (L () ), for which / = hm ((/ - T)h n + (XI - h) (I - T)g n ). n—KX) Empowering People. Improving Business. Norwegian Business School is one of Europe's largest business schools welcoming more than 20,000 students. Our programmes provide a stirnufating and multi-cultural learning environment with an international outlook ultimately providing students with professional skills to meet the increasing needs of businesses. B! offers four different two-yea i; full-time Master of Science (MSc) programmes that are taught entirely in English and have been designed to provide professional skills to meet the increasing need of businesses.The MSc piiogramme5 provide a stimulating and multi¬ cultural learning environment to give you the best platform to launch into your career * MSc in Business * MSc in Financial Economics * MSc in Strategic Marketing Management NORWEGIAN BUSINESS SCHOOL EFMD EQUIS *ffi * MSc in Leadership and Organisational Psychology www.bi.edu/master 233 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Put /„ = (/- T)K + (A/ - L 0 (/ - T)g n . Then (a/ - Z) G(A)/ n = f n (see (3.332) and (3.333)). Since the operator L verifies the maximum principle it follows that ARe G(X)f n ^ inf Re (xi - L ) G(X)f n (y ) = inf Re f n (y ) (3.336) and hence Re XG(X)f = Re lim„^, x , XG(X)f n ^ 0. A similar argument will show that the operator G( A) sends real functions to real function and hence G(X) is positivity preserving. Next we prove that the family {G(X) : A > 0, large} is a resolvent family. So let A and g be large positive real numbers. We want to prove the identity G(X) - G(g) -(g- X)G(g)G(X) = 0. (3.337) First pick the function / 6 D(L 0 ) and apply the operator in (3.337) to the function (A I — L{) (/ — T)f and employ identity G( A) ^A I — L'j f = f, for / belonging to D (^L^j to obtain (G(X) - G(g) -(g- X)G(g)G{ A)) (XI - L x ) (I - T)f = 0. The operator in (3.337) also sends functions in the space R(I — T) to 0, because we may apply (3.333) to see that (l-il ~ Z) (G(X) - G(g) - (g - X)G(g)G(X)) (I -T)f = 0 for / 6 D(L 0 ). Finally we show that the resolvent family (G'(A) : A > 0 large} is strongly continuous in the sense that lim^oo XR(X)f = f for all / e Cq(E). Of course it suffices to prove this equality for a subset with a dense span. Next we consider / e D(L 0 ) and we estimate ||(/-r)/-AG(A)(/-T)/|| co as follows: |(/-T)/-AG(A)(/-T)/|L « } |(A/ - h) ((I - T)f - \G(\)(I - T)/)||„ - } |(A/ - L r ) (1 - T)f - A(J - T)/||„ = } ||L,(/ - T)f\\ x . (3.338) Again this expression tends to zero. For brevity we write For / e D(Lq) the following equalities are valid: (XG(X)-I)L l (I-T)f-L l (I-T)f = A 2 G(A)(I - T)f - X(I - T)f — L\(I — T)f = {A 2 (/ - T)R(X) + TV(X)R(X) 2 - X(I - T)(I - V(X)R(X))} F( A) — L\(I — T)f = {(/ - T)XL 0 R(X ) + X 2 TV(X)R(X) 2 + X (/ - T)V(X)R(X)} F( A) 234 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales -u (I-T)f -» {(/ - T)Lq +TV + (I - T)Vj T)f = 0. (3.339) From (3.338) together with (3.339) we conclude that lim^oo ( XG(X)f — /) = 0 for all in the span of R(I — T ) and R (L X (I — T)). By assumption this span is dense and consequently the resolvent family {G( A) : A > 0, A large} is strongly continuous. In order to conclude the proof of the existence result we choose f x and f 2 in the space D(Lq) and we notice the following identities: G(A) {(/ - T) (I - V(X)R(X)) (XI - L 0 )f x + (XI - L X )(I - T)f 2 } = (I- T)(h + h) + TV(\)R(\)f u and so the space G(X)Co(E) contains the linear span of the spaces R(I — T ) and R(TV(X)R(X)). From the resolvent equation it is clear that the space G(X)Cq(E ) does not depend on the variable A. So we see that the space G(a)Co(E) contains, for a given function / 6 D(L 0 ) the family {XTV(X)R(X)f : X ^ a}. Hence the function TVf = lim XTV(X)R(X)f belongs to the closure of A—»oo the space G(a)C 0 (E). Since L x (I-T)f = TVf + (I-T) (L x (I - T)f + TL 0 f), for / 6 D(L 0 ), we conclude that the range of L\(I — T) is contained in the clo¬ sure of G(a)Co(E). Since the latter space also contains R(I — T) it follows from the density of the space R(I-T) + R(L 1 (I-T)) in Cq(E) that the domain of the resolvent, i.e. G(o)Cq(E ) is dense in C$(E). From the previous discussion it also follows that the operator which assigns to G(X)f the function XG(X)f — f extends the operator Li restricted to R(I — T). It is now also clear that the subspace {G(a)f : f e Co(E)} is dense and so it is clear that the there exists a Feller semigroup generated by the operator L with graph {(G(a)f, aG(a)f - f) : f e C 0 (E)}. For the uniqueness we proceed as follows. Let P} and P^ : be two solutions for the martingale problem. We define the family of operators {S(t) : t V 0} as follows: S(t)f(x ) = E lf(X(t) — Elf(X(t)) from the martingale property, it then follows that S'(t)f = S(t)Lf for / belonging to the subspace R(I — T) + R(L\(I — T)). Moreover we have S(0)f(x) = 0 for all functions / e Cq(E). Then we write (for A and f 2 e D(L 0 )) -f -f -f ~ ) e~ Xt S(t)G( A) ((/ - T)A + TL x (I - T)f 2 ) dt e~ Xt S(t) [XI-L) G( A) ((I - T)fi + TL X (I - T)f 2 ) dt e~ xt S(t ) ((I - T)f x + TL X (I - T)f 2 ) dt. (3.340) Consequently S(t)(I — T)f x = S(t)TL x (I — T)f 2 = 0 for all functions f x and f n in the space D(Lq). We also have, upon using (3.340) the following equality: f S(t)L x (I - T)fdr = S(t)(I - T)f - S(0)f = 0. (3.341) 235 Download free eBooks at bookboon.com Advanced stochastic processes: Part I An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales Since by assumption the sum of the vector spaces R(I — T ) and R(Li) is dense in the space Cq(E ), we conclude S(t) = 0 and hence from a general result on uniqueness of the martingale problem, we finally obtain that P* = P^ for all x e E. For more details see Proposition 2.9 (Corollary p. 206 of Ikeda and Watanabe [61]). This completes the proof of Theorem 3.110. □ 236 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Index Index D : dyadic rational numbers, 380 K: strike price, 191 N(-): normal distribution, 191 P'(Cl): compact metrizable Hausdorff space, 129 S: spot price, 191 T: maturity time, 191 A-system, 1, 68 S( 5 -set, 332, 334 M: space of complex measures on M*', 298 Vo’l 103 7r-system, 68 cr-algebra, 1, 3 cr-field, 1, 3 a: volatility, 191 r: risk free interest rate, 191 (DL)-property, 416 adapted process, 17, 374, 389, 406 additive process, 23, 24 affine function, 8 affine term structure model, 210 Alexandroff compactification, 301 almost sure convergence of sub-martingales, 386 arbitrage-free, 190 backward propagator, 197 Banach algebra, 298, 303 Bernoulli distributed random variable, 56 Bernoulli topology, 310 Beurling-Gelfand formula, 302, 303 Birkhoff’s ergodic theorem, 74 birth-dearth process, 35 Black-Scholes model, 187, 190 Black-Scholes parameters, 193 Black-Scholes PDE, 190 Bochner’s theorem, 90, 91, 308, 314 Boolean algebra of subsets, 361 Borel-Cantelli’s lemma, 42, 105 Brownian bridge, 94, 98, 99, 101 Brownian bridge measure conditional, 103 Brownian motion, 1, 16-18, 24, 84, 94, 98, 101, 102, 105, 108-110, 113, 115, 181, 189, 193, 197, 243, 283, 290, 291 continuous, 104 distribution of, 107 geometric, 188 Holder continuity of, 154 pinned, 98 standard, 70 Brownian motion with drift, 98 cadlag modification, 395 cadlag process, 376 Cameron-Mart in Girsanov formula, 277 Cameron-Martin transformation, 182, 280 canonical process, 109 Caratheodory measurable set, 363 Caratheodory’s extension theorem, 361, 362, 364 central limit theorem, 74 multivariate, 70 Chapman-Kolmogorov identity, 16, 25, 81, 107, 116, 149 characteristic function, 76, 102, 390 characteristic function (Fourier transform), 98 classification properties of Markov chains, 35 closed martingale, 17, 150 compact-open topology, 310 complex Radon measure, 296 conditional Brownian bridge measure, 103 conditional expectation, 2, 3, 78 conditional expectation as orthogonal projection, 5 conditional expectation as projection, 5 conditional probability kernel, 399 consistent family of probability spaces, 66 consistent system of probability measures, 13, 360 content, 362 exended, 362 continuity theorem of Levy, 324 237 Download free eBooks at bookboon.com Index Advanced stochastic processes: Part I contractive operator, 197 convergence in probability, 371, 386 convex function and affine functions, 8 convolution product of measures, 298 convolution semigroup of measures, 314 convolution semigroup of probability measures, 391 coupling argument, 288 covariance matrix, 108, 197, 200, 203 cylinder measure, 360 cylinder set, 358, 367 cylindrical measure, 89, 125 decomposition theorem of Doob-Meyer, 20 delta hedge portfolio, 190 density function, 80 Dirichlet problem, 265 discounted pay-off, 209 discrete state space, 25 discrete stopping time, 19 dispersion matrix, 94 dissipative operator, 118 distribution of random variable, 102 distributional solution, 266 Doleans measure, 168 Donsker’s invariance principle, 71 Doob’s convergence theorem, 17, 18 Doob’s maximal inequality, 21, 23, 160, 384 Doob’s maximality theorem, 21 Doob’s optional sampling theorem, 20, 86, 381, 388, 409 Doob-Meyer decomposition for discrete sub-martingales, 383 Doob-Meyer decomposition theorem, 148, 149, 295, 384, 410, 419, 421 downcrossing, 157 Dynkin system, 1, 68, 111, 300, 378 Elementary renewal theorem, 38 equi-integrable family, 369 ergodic theorem, 295 ergodic theorem in L 2 , 342 ergodic theorem of Birkhoff, 76, 340, 344, 354 European call option, 188 European put option, 188 event, 1 exit time, 84 exponential Brownian motion, 186 exponential local martingale, 254, 255 exponential martingale probability measure, 192 extended content, 362 extension theorem of Kolmogorov, 360 exterior measure, 364 face value, 210 Feller semigroup, 79, 113, 114, 120, 121, 140, 264 conservative, 114 generator of, 118, 137, 140, 143, 144 strongly continuous, 113 Feller-Dynkin semigroup, 79, 122, 264 Feynman-Kac formula, 181 filtration, 109, 264 right closure of, 109 finite partition, 3 finite-dimensional distribution, 373 first hitting time, 18 forward propagator, 197 forward rate, 214 Fourier transform, 90, 93, 96, 102, 251 Fubini’s theorem, 199 full history, 109 function positive-definite, 305 functional central limit theorem (FCLT), 70, 71 Gaussian kernel, 16, 107 Gaussian process, 89, 110, 115, 200, 203 Gaussian variable, 153 Gaussian vector, 76, 93, 94 GBM, 186 geometric Brownian motion, 189 generator of Feller semigroup, 118, 137, 140, 144, 228, 230, 231, 233 generator of Markov process, 200, 203 geometric Brownian motion, 188 geometric Brownian motion = GBM, 186 Girsanov transformation, 182, 243, 280 Girsanov’s theorem, 193 graph, 232 Gronwall’s inequality, 246 Holder continuity of Brownian motion, 154 Holder continuity of processes, 151 Hahn decomposition, 295 Hahn-Kolmogorov’s extension theorem, 364 harmonic function, 86 hedging strategy, 188 Hermite polynomial, 258 Hilbert cube, 333, 334 hitting time, 18 i.i.d. random variables, 24 238 Download free eBooks at bookboon.com Index Advanced stochastic processes: Part I index set, 11 indistinguishable processes, 104, 374, 386 information from the future, 374 initial reward, 40 integration by parts formula, 282 interest rate model, 204 internal history, 374, 394 invariant measure, 35, 48, 51, 201, 204 minimal, 50 irreducible Markov chain, 48, 51, 54 Ito calculus, 87, 278, 279 Ito isometry, 162 ltd representation theorem, 274 Ito’s lemma, 189, 270 Jensen inequality, 149 Kolmogorov backward equation, 26 Kolmogorov forward equation, 26 Kolmogorov matrix, 26 Kolmogorov’s extension theorem, 13, 17, 89-91, 93, 125, 130, 357, 360, 361, 366 Komlos’ theorem, 295, 409, 420 Levy’s weak convergence theorem, 115 Levy process, 89, 389, 390, 392 Levy’s characterization of Brownian motion, 194, 249 law of random variable, 102 Lebesgue-Stieltjes measure, 364 lemma of Borel-Cantelli, 10, 152 lexicographical ordering, 333 life time, 79, 117 local martingale, 194, 252, 264, 267, 268, 271, 278, 280 local time, 292 locally compact Hausdorff space, 15 marginal distribution, 373 marginal of process, 13 Markov chain, 35, 44, 58, 59, 66 irreducible, 48, 54 recurrent, 48 Markov chain recurrent, 48 Markov process, 1, 16, 29, 30, 61, 79, 89, 102, 110, 113, 115, 119, 144, 202, 406, 408 strong, 119, 406 time-homogeneous, 407 Markov property, 25, 26, 30, 31, 46, 82, 110, 113, 142 strong, 44 martingale, 1, 17, 20, 80-82, 85-88, 103, 109, 243, 280, 281, 378, 382, 396 (DL)-property, 227 closed, 17 local, 194 maximal inequality for, 225 martingale measure, 209, 281 martingale problem, 118, 128, 137, 140, 143, 144, 228, 230, 231, 235, 264, 265 uniquely solvable, 118 well-posed, 118 martingale property, 131 martingale representation theorem, 263, 275 maximal ergodic theorem, 351 maximal inequality of Doob, 386 maximal inequality of Levy, 104 maximal martingale inequality, 225 maximum principle, 118, 140, 141, 143, 232 measurable mapping, 377 measure invariant, 48, 201, 204 mesaure invariant, 204 mesure stationary, 204 metrizable space, 15 Meyer process, 419 minimal invariant measure, 50 modification, 374 monotone class theorem, 69, 103, 107, 110, 112, 116, 378, 394, 398, 401, 404 alternative, 378 multiplicative process, 23, 24, 79 multivariate classical central limit theorem, 70 multivariate normal distributed vector, 76 multivariate normally distributed random vector, 93 negative-definite function, 314, 316, 396 no-arbitrage assumption, 209 non-null recurrent state, 51 non-null state, 47 non-positive recurrent random walk, 57 non-time-homogeneous process, 23 normal cumulative distribution, 188 normal distribution, 197 Novikov condition, 281 Novikov’s condition, 209 null state, 47 numeraire, 215 number of upcrossings, 156, 379, 380 one-point compactification, 15 operator 239 Download free eBooks at bookboon.com Index Advanced stochastic processes: Part I dissipative, 118 operator which maximally solves the martingale problem, 118, 140, 228 Ornstein-Uhlenbeck process, 98, 102, 200, 201, 210 orthogonal projection, 340 oscillator process, 98, 99 outer measure, 363, 364 partial reward, 40 partition, 4 path, 373 path space, 117 pathwise solutions to SDE, 288, 289 unique, 291, 292 pathwise solutions to SDE’s, 244 payoff process discounted, 193 PDE for bond price in the Vasicek model, 213 pe-measure, 362 persistent state, 47 pinned Brownian motion, 98 Poisson process, 26, 27, 29, 36, 89, 159 Polish space, 15, 90, 123, 334, 335, 360, 361, 366 portfolio delta hedge, 190 positive state, 47 positive-definite function, 297, 302, 305, 314 positive-definite matrix, 90, 96, 197 positivity preserving operators, 345 pre-measure, 363, 364 predictable process, 20, 193, 418 probability kernel, 399, 408 probability measure, 1 probability space, 1 process Gaussian, 200, 203 increasing, 21 predictable, 20 process adapted to filtration, 374 process of class (DL), 20, 21, 148, 149, 161, 409-411, 420, 421 progressively measurable process, 377 Prohorov set, 72, 335, 337-339 projective system of probability measures, 13, 121, 360 projective system of probability spaces, 125 propagator backward, 197 quadratic covariation process, 249, 264, 279 quadratic variation process, 253 Radon-Nikodym derivative, 11, 408 Radon-Nikodym theorem, 4, 78, 408 random walk, 58 realization, 25, 373 recurrent Markov chain, 48 recurrent state, 47 recurrent symmetric random walk, 55 reference measure, 80, 81, 83 reflected Brownian motion, 228 renewal function, 35 renewal process, 35, 40 renewal-reward process, 39, 40 renewal-reward theorem, 41 resolvent family, 122 return time, 55 reward initial, 40 partial, 40 terminal, 40 reward function, 40 Riemann-Stieltjes integral, 364 Riesz representation theorem, 295, 296, 305 right closure of filtration, 109 right-continuous filtration, 374 right-continuous paths, 19 ring of subsets, 361 risk-neutral measure, 193, 209 risk-neutral probability measure, 192 running maximum, 23 sample path, 25 sample path space, 11, 25 sample space, 25 semi-martingale, 419 semi-ring, 364 semi-ring of subsets, 361, 362 semigroup Feller, 264 Feller-Dynkin, 264 shift operator, 109, 117 Skorohod space, 117, 122, 128 Skorohod-Dudley-Wichura representation theorem, 283, 286 Souslin space, 90, 361, 365, 366 space-homogeneous process, 29 spectral radius, 303 standard Brownian motion, 70 state non-null, 47 null, 47 240 Download free eBooks at bookboon.com Index Advanced stochastic processes: Part I persistent, 47 positive, 47 recurrent, 47 state space, 11, 17, 79, 117, 400, 406 discrete, 25 state variable, 11, 25, 117 state variables, 125 stateitransient, 47 stationary distribution, 25, 51 stationary measure, 204 stationary process, 11 step functions with unit jumps, 159 Stieltjes measure, 364 Stirling’s formula, 54 stochastic differential equation, 182 stochastic integral, 102, 253 stochastic process, 10 stochastic variable, 11, 371 stochastically continuous process, 159 stochastically equivalent processes, 374 stopped filtration, 377 stopping time, 18, 20, 44, 58, 64, 68, 112, 252, 374-377, 381, 382, 405 discrete, 19 terminal, 18, 24 strong law of large numbers, 41, 76, 155, 340, 344 strong law of large numbers (SLLN), 38 strong Markov process, 102, 119, 121, 140, 406 strong Markov property, 44, 48, 113 strong solution to SDE, 244 strong solutions to SDE unique, 244 strong time-dependent Markov property, 113, 120 strongly continuous Feller semigroup, 113 sub-martingale, 378, 381, 384 sub-probability kernel, 406 sub-probability measure, 1 submartingale, 17, 20, 227 submartingale convergence theorem, 158 submartingale of class (DL), 421 super-martingale, 378 supermartingale, 17, 20 Tanaka’s example, 292 terminal reward, 40 terminal stopping time, 18, 24, 83 theorem Ito representation, 274 Kolmogorov’s extension, 278 martingale representation, 275 of Arzela-Ascoli, 72, 73 of Bochner, 90, 304, 308 of Doob-Meyer, 20 of Dynkin-Hunt, 397 of Fernique, 221 of Fubini, 199, 330 of Girsanov, 277, 280 of Helly, 334 of Komlos, 409 of Levy, 253, 270, 290 of Prohorov, 72 of Radon-Nikodym, 290 of Riemann-Lebesgue, 300 of Scheffe, 39, 278, 369 of Schoenberg, 314 of Stone-Weierstrass, 301, 305 S kor oho d- D udley-Wichur a representation, 283, 286 time, 11 time change, 19 stochastic, 19 time-dependent Markov process, 200, 203 time-homogeneous process, 11, 29 time-homogeneous transition probability, 25 time-homogenous Markov process, 407 topology of uniform convergence on compact subsets, 310 tower property of conditional expectation, 5 transient non-symmetric random walk, 57 transient state, 47 transient symmetric random walk, 55 transition function, 119 transition matrix, 51 translation operator, 11, 25, 109, 117, 400, 406 translation variables, 125 uniformly distributed random variable, 394 uniformly integrable family, 5, 6, 20, 39, 369, 388 uniformly integrable martingale, 389 uniformly integrable sequence, 385 unique pathwise solutions to SDE, 244 uniqueness of the Doob-Meyer decomposition, 417 unitary operator, 340, 342 upcrossing inequality, 156, 157, 383 upcrossing times, 156 upcrossings, 156 vague convergence, 371 vague topology, 310, 334 vaguely continuous convolution semigroup of measures, 315 241 Download free eBooks at bookboon.com Advanced stochastic processes: Part I Index vaguely continuous convolution semigroup of probability measures, 389, 390 Vasicek model, 204, 210 volatility, 188 von Neumann’s ergodic theorem, 340 Wald’s equation, 36 weak convergence, 325 weak law of large numbers, 75, 340 weak solutions, 264 weak solutions to SDE’s, 244, 277, 280, 288 unique, 265, 292 weak solutions to stochastic differential equations, 265 weak topology, 310 weak*-topology, 334 weakly compact set, 338, 339 Wiener process, 98 Brain power By 2020, wind could provide one-tenth of our planet’s 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